Theorem ialgrlemconst 12078

Description: Lemma for ialgr0 12079. Closure of a constant function, in a form suitable for theorems such as seq3-1 10493 or seqf 10494. (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 2256	. . . 4 |- ( x e. Z <-> x e. ( ZZ>= ` M ) )
4	3	biimpri 133	. . 3 |- ( x e. ( ZZ>= ` M ) -> x e. Z )
5		fvconst2g 5751	. . 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 2266	1 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   {csn 3607   X. cxp 4642   ` cfv 5235   ZZ>=cuz 9559

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243

This theorem is referenced by:  ialgr0  12079  algrp1  12081  mulgnn0z  13106  mulgnndir  13108  mulgpropdg  13121