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WLPO implies Decidability of equality with the point at infinity in NN+oo #4469

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merged 2 commits into from
Dec 8, 2024
Merged

WLPO implies Decidability of equality with the point at infinity in NN+oo #4469

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abd9dff
copy fnmptfvd from set.mm to iset.mm
jkingdon Dec 6, 2024
f7f3602
Add nninfdcinf to iset.mm
jkingdon Dec 4, 2024
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36 changes: 36 additions & 0 deletions iset.mm
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Original file line number Diff line number Diff line change
Expand Up @@ -50496,6 +50496,25 @@ We use their notation ("onto" under the arrow). (Contributed by NM,
CUHUJTVAUQACUNCRUSUOUTUPUNCDSUNCESUAUBUCUD $.
$}

${
$d A a i $. $d C i $. $d D a $. $d M a i $. $d U a i $. $d V a i $.
$d ph a i $.
fnmptfvd.m $e |- ( ph -> M Fn A ) $.
fnmptfvd.s $e |- ( i = a -> D = C ) $.
fnmptfvd.d $e |- ( ( ph /\ i e. A ) -> D e. U ) $.
fnmptfvd.c $e |- ( ( ph /\ a e. A ) -> C e. V ) $.
$( A function with a given domain is a mapping defined by its function
values. (Contributed by AV, 1-Mar-2019.) $)
fnmptfvd $p |- ( ph -> ( M = ( a e. A |-> C )
<-> A. i e. A ( M ` i ) = D ) ) $=
( cmpt wceq cfv wral wfn wcel eqid cv ralrimiva fnmpt syl syl2anc cbvmptv
wb eqfnfv eqcomi a1i fveq1d eqeq2d ralbidv simpr fvmpt2 ralbidva 3bitrd
wa ) AGIBCNZOZFUAZGPZVAUSPZOZFBQZVBVAFBDNZPZOZFBQVBDOZFBQAGBRUSBRZUTVEUGJ
ACHSZIBQVJAVKIBMUBIBCUSHUSTUCUDFBGUSUHUEAVDVHFBAVCVGVBAVAUSVFUSVFOAVFUSFI
BDCKUFUIUJUKULUMAVHVIFBAVABSZURZVGDVBVMVLDESVGDOAVLUNLFBDEVFVFTUOUEULUPUQ
$.
$}

${
$d F x y $. $d G x y $. $d A x y $.
$( Two ways to express the locus of differences between two functions.
Expand Down Expand Up @@ -71014,6 +71033,23 @@ of Omniscience (WLPO).
( cen wbr cwomni wcel enwomnilem wi ensym syl impbid ) ABCDZAEFZBEFZABGLBAC
DNMHABIBAGJK $.

${
$d N f x $. $d N i x $. $d i ph x $.
nninfdcinf.w $e |- ( ph -> _om e. WOmni ) $.
nninfdcinf.n $e |- ( ph -> N e. NN+oo ) $.
$( The Weak Limited Principle of Omniscience (WLPO) implies that it is
decidable whether an element of ` NN+oo ` equals the point at infinity.
(Contributed by Jim Kingdon, 3-Dec-2024.) $)
nninfdcinf $p |- ( ph -> DECID N = ( i e. _om |-> 1o ) ) $=
( vx vf com c1o wceq wdc cv cfv wral c2o dcbid cwomni wcel cvv syl eqeq1d
cmpt cmap co fveq1 ralbidv wb elexd iswomnimap mpbid wf xnninf 2onn elexi
nninff omex elmap sylibr rspcdva ffnd eqidd wa 1onn a1i fnmptfvd mpbird )
ACBHIUBJZKFLZCMZIJZFHNZKZAVHGLZMZIJZFHNZKZVLGOHUCUDZCVMCJZVPVKVSVOVJFHVSV
NVIIVHVMCUEUAUFPAHQRZVQGVRNZDAHSRVTWAUGAHQDUHFHGSUITUJAHOCUKZCVRRACULRWBE
CUOTZOHCOHUMUNUPUQURUSAVGVKAHIIHFCHBAHOCWCUTVHBLZJIVAIHRZAVHHRVBVCVDWEAWD
HRVBVCVDVEPVF $.
$}


$(
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