Theorem ialgrlemconst 12435

Description: Lemma for ialgr0 12436. Closure of a constant function, in a form suitable for theorems such as seq3-1 10624 or seqf 10626. (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 2273	. . . 4 |- ( x e. Z <-> x e. ( ZZ>= ` M ) )
4	3	biimpri 133	. . 3 |- ( x e. ( ZZ>= ` M ) -> x e. Z )
5		fvconst2g 5810	. . 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 2283	1 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   {csn 3637   X. cxp 4680   ` cfv 5279   ZZ>=cuz 9663

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 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-pr 4260

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 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-fv 5287

This theorem is referenced by:  ialgr0  12436  algrp1  12438  mulgnn0z  13555  mulgnndir  13557  mulgpropdg  13570