Theorem ialgrlemconst 12620

Description: Lemma for ialgr0 12621. Closure of a constant function, in a form suitable for theorems such as seq3-1 10725 or seqf 10727. (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 2298	. . . 4 |- ( x e. Z <-> x e. ( ZZ>= ` M ) )
4	3	biimpri 133	. . 3 |- ( x e. ( ZZ>= ` M ) -> x e. Z )
5		fvconst2g 5868	. . 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 2308	1 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   {csn 3669   X. cxp 4723   ` cfv 5326   ZZ>=cuz 9755

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334

This theorem is referenced by:  ialgr0  12621  algrp1  12623  mulgnn0z  13741  mulgnndir  13743  mulgpropdg  13756