Theorem ialgrlemconst 12480

Description: Lemma for ialgr0 12481. Closure of a constant function, in a form suitable for theorems such as seq3-1 10644 or seqf 10646. (Contributed by Jim Kingdon, 22-Jul-2021.)

Hypotheses

Ref	Expression
ialgrlemconst.z	|- Z = ( ZZ>= ` M )
ialgrlemconst.a	|- ( ph -> A e. S )

Assertion

Ref	Expression
ialgrlemconst	|- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) e. S )

Proof of Theorem ialgrlemconst

Step	Hyp	Ref	Expression
1		ialgrlemconst.a	. . 3 |- ( ph -> A e. S )
2		ialgrlemconst.z	. . . . 5 |- Z = ( ZZ>= ` M )
3	2	eleq2i 2274	. . . 4 |- ( x e. Z <-> x e. ( ZZ>= ` M ) )
4	3	biimpri 133	. . 3 |- ( x e. ( ZZ>= ` M ) -> x e. Z )
5		fvconst2g 5821	. . 3 |- ( ( A e. S /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) = A )
6	1, 4, 5	syl2an 289	. 2 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) = A )
7	1	adantr 276	. 2 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A e. S )
8	6, 7	eqeltrd 2284	1 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   {csn 3643   X. cxp 4691   ` cfv 5290   ZZ>=cuz 9683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298

This theorem is referenced by:  ialgr0  12481  algrp1  12483  mulgnn0z  13600  mulgnndir  13602  mulgpropdg  13615