Theorem ialgrlemconst 11997

Description: Lemma for ialgr0 11998. Closure of a constant function, in a form suitable for theorems such as seq3-1 10416 or seqf 10417. (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 2237	. . . 4 |- ( x e. Z <-> x e. ( ZZ>= ` M ) )
4	3	biimpri 132	. . 3 |- ( x e. ( ZZ>= ` M ) -> x e. Z )
5		fvconst2g 5710	. . 3 |- ( ( A e. S /\ x e. Z ) -> ( ( Z X. { A } ) ` x ) = A )
6	1, 4, 5	syl2an 287	. 2 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) = A )
7	1	adantr 274	. 2 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A e. S )
8	6, 7	eqeltrd 2247	1 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) e. S )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   {csn 3583   X. cxp 4609   ` cfv 5198   ZZ>=cuz 9487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206

This theorem is referenced by:  ialgr0  11998  algrp1  12000