Theorem mhmlem 17535

Description: Lemma for mhmmnd 17537 and ghmgrp 17539. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)

Hypotheses

Ref	Expression
ghmgrp.f	|- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
mhmlem.a	|- ( ph -> A e. X )
mhmlem.b	|- ( ph -> B e. X )

Assertion

Ref	Expression
mhmlem	|- ( ph -> ( F ` ( A .+ B ) ) = ( ( F ` A ) .+^ ( F ` B ) ) )

Distinct variable groups:   x, F, y   x, .+ , y   x, X, y   x, .+^ , y   ph, x, y   x, A, y   y, B

Allowed substitution hint:   B( x)

Proof of Theorem mhmlem

Step	Hyp	Ref	Expression
1		id 22	. 2 |- ( ph -> ph )
2		mhmlem.a	. 2 |- ( ph -> A e. X )
3		mhmlem.b	. 2 |- ( ph -> B e. X )
4		eleq1 2689	. . . . . 6 |- ( x = A -> ( x e. X <-> A e. X ) )
5	4	3anbi2d 1404	. . . . 5 |- ( x = A -> ( ( ph /\ x e. X /\ y e. X ) <-> ( ph /\ A e. X /\ y e. X ) ) )
6		oveq1 6657	. . . . . . 7 |- ( x = A -> ( x .+ y ) = ( A .+ y ) )
7	6	fveq2d 6195	. . . . . 6 |- ( x = A -> ( F ` ( x .+ y ) ) = ( F ` ( A .+ y ) ) )
8		fveq2 6191	. . . . . . 7 |- ( x = A -> ( F ` x ) = ( F ` A ) )
9	8	oveq1d 6665	. . . . . 6 |- ( x = A -> ( ( F ` x ) .+^ ( F ` y ) ) = ( ( F ` A ) .+^ ( F ` y ) ) )
10	7, 9	eqeq12d 2637	. . . . 5 |- ( x = A -> ( ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) <-> ( F ` ( A .+ y ) ) = ( ( F ` A ) .+^ ( F ` y ) ) ) )
11	5, 10	imbi12d 334	. . . 4 |- ( x = A -> ( ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) ) <-> ( ( ph /\ A e. X /\ y e. X ) -> ( F ` ( A .+ y ) ) = ( ( F ` A ) .+^ ( F ` y ) ) ) ) )
12		eleq1 2689	. . . . . 6 |- ( y = B -> ( y e. X <-> B e. X ) )
13	12	3anbi3d 1405	. . . . 5 |- ( y = B -> ( ( ph /\ A e. X /\ y e. X ) <-> ( ph /\ A e. X /\ B e. X ) ) )
14		oveq2 6658	. . . . . . 7 |- ( y = B -> ( A .+ y ) = ( A .+ B ) )
15	14	fveq2d 6195	. . . . . 6 |- ( y = B -> ( F ` ( A .+ y ) ) = ( F ` ( A .+ B ) ) )
16		fveq2 6191	. . . . . . 7 |- ( y = B -> ( F ` y ) = ( F ` B ) )
17	16	oveq2d 6666	. . . . . 6 |- ( y = B -> ( ( F ` A ) .+^ ( F ` y ) ) = ( ( F ` A ) .+^ ( F ` B ) ) )
18	15, 17	eqeq12d 2637	. . . . 5 |- ( y = B -> ( ( F ` ( A .+ y ) ) = ( ( F ` A ) .+^ ( F ` y ) ) <-> ( F ` ( A .+ B ) ) = ( ( F ` A ) .+^ ( F ` B ) ) ) )
19	13, 18	imbi12d 334	. . . 4 |- ( y = B -> ( ( ( ph /\ A e. X /\ y e. X ) -> ( F ` ( A .+ y ) ) = ( ( F ` A ) .+^ ( F ` y ) ) ) <-> ( ( ph /\ A e. X /\ B e. X ) -> ( F ` ( A .+ B ) ) = ( ( F ` A ) .+^ ( F ` B ) ) ) ) )
20		ghmgrp.f	. . . 4 |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
21	11, 19, 20	vtocl2g 3270	. . 3 |- ( ( A e. X /\ B e. X ) -> ( ( ph /\ A e. X /\ B e. X ) -> ( F ` ( A .+ B ) ) = ( ( F ` A ) .+^ ( F ` B ) ) ) )
22	2, 3, 21	syl2anc 693	. 2 |- ( ph -> ( ( ph /\ A e. X /\ B e. X ) -> ( F ` ( A .+ B ) ) = ( ( F ` A ) .+^ ( F ` B ) ) ) )
23	1, 2, 3, 22	mp3and 1427	1 |- ( ph -> ( F ` ( A .+ B ) ) = ( ( F ` A ) .+^ ( F ` B ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  mhmid  17536  mhmmnd  17537  ghmgrp  17539  ghmcmn  18237