Add imadomnum, and drop the ax-ac dependency from fimact - #5466
Open
vince-gonzalez wants to merge 2 commits into
Open
Add imadomnum, and drop the ax-ac dependency from fimact#5466vince-gonzalez wants to merge 2 commits into
vince-gonzalez wants to merge 2 commits into
Conversation
set.mm states dominance of an image by its index set three times and gives the well-orderable-domain version of only two: fodomg has fodomnum, and fnrndomg has fnrndomnum since metamath#5443. imadomg had none. imadomnum is that one, in imadomg's own shape with A e. B weakened to A e. dom card. It is proved from fodomnum rather than fnrndomnum on purpose. fnrndomnum sits six lines below fimact, so building on it would force one of the two statements to move; fodomnum is above both, and fnrndomnum is itself only dffn4 then fodomnum. Nothing moves. fimact holds A ~<_ _om, so ondomen gives A e. dom card and imadomnum applies. 71 steps to 108. fimactOLD keeps the previous proof, following fnctOLD and dmctOLD; discouraged regenerated, two lines added. The count does not change: 473 before and after, because fimactOLD still reaches the axiom through imadomg.
avekens
reviewed
Aug 28, 2026
Requested by avekens on the pull request. imadomnum carried one and fimact did not, which was inconsistent. Written as avoids 'ax-ac' 'ax-ac2' rather than 'ax-ac' alone: fimact avoids both, and that is the form 35 existing guards in the file use.
avekens
approved these changes
Aug 28, 2026
tirix
approved these changes
Aug 28, 2026
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
This frees nothing on its own, and with
fimactOLDkept it does not move thecount at all: 473 statements reach
ax-ac/ax-ac2before and after, becausethe old proof retained under
fimactOLDstill goes throughimadomg. Samearithmetic as #5458, where keeping the two OLD copies took 55 down to 53.
It is worth having for the lemma.
imadomnum
set.mm states dominance of an image by its index set three times and gives the
well-orderable-domain version of only two:
fodomgfodomnumfnrndomgfnrndomnum(#5443)imadomgimadomnumis the missing one, stated in exactlyimadomg's shape withA e. Bweakened toA e. dom card, the same wayfnrndomnumrelates tofnrndomg.Two things about it are deliberate.
It goes through
fodomnum, notfnrndomnum.fnrndomnumis at line118104 of develop and
fimactat 118098, six lines above it, so animadomnumproved fromfnrndomnumcould not be placed anywherefimactcould cite it and one of the two would have to move.
fodomnumis at 109793,before both. Since
fnrndomnumis itself onlydffn4thenfodomnum, goingstraight to
fodomnumcosts nothing and moves no statement.It is curried rather than conjoined. Its two siblings in the main part are
curried, and the conjoined precedent,
imadomfi, is in a mathbox and so cannotbe named from here.
The route never needs a surjection of its own. Restricting
FtoAgives afunction on
( A i^i dom F ); that set is a subset of a well-orderable setand so well-orderable by
ssnum;dffn4turns theFninto an onto andfodomnumbounds the range. Rewritingran ( F |A )as( F " A )and composing with( A i^i dom F ) ~<_ A ` finishes it.fimact
The first caller.
A ~<_ _omis a hypothesis, soomelonandondomengiveA e. dom cardandimadomnumapplies whereimadomgwas used. 71 steps to108.
fimactOLDkeeps the previous proof, followingfnctOLDanddmctOLDfrom#5458.
discouragedregenerated: two lines added, none removed.Why it is worth landing anyway
fimactis one of four remaining callers of these three lemmas that hold ahypothesis making their own domain well-orderable. Measured together the four
free 134 statements; measured one at a time they free 60, 49, 1 and 1. Where
several of them feed one subtree, each reads near zero alone and the subtree
only leaves the cone when the last one goes.
fimactis one of the ones thatreads as 1.
imadomnumis also the lemma the other three need in order to be statedwithout choice, in the same way #5443 froze nothing by itself.
Checks run
verify proof *on the whole database, no errorsverify markup *, no errorsscripts/rewrapapplied, so the file is in canonical formscripts/regen-discouraged, two lines added forfimactOLD