Theorem mhmlem 13450

Description: Lemma for mhmmnd 13452 and ghmgrp 13454. (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 19	. 2 |- ( ph -> ph )
2		mhmlem.a	. 2 |- ( ph -> A e. X )
3		mhmlem.b	. 2 |- ( ph -> B e. X )
4		eleq1 2268	. . . . . 6 |- ( x = A -> ( x e. X <-> A e. X ) )
5	4	3anbi2d 1330	. . . . 5 |- ( x = A -> ( ( ph /\ x e. X /\ y e. X ) <-> ( ph /\ A e. X /\ y e. X ) ) )
6		fvoveq1 5967	. . . . . 6 |- ( x = A -> ( F ` ( x .+ y ) ) = ( F ` ( A .+ y ) ) )
7		fveq2 5576	. . . . . . 7 |- ( x = A -> ( F ` x ) = ( F ` A ) )
8	7	oveq1d 5959	. . . . . 6 |- ( x = A -> ( ( F ` x ) .+^ ( F ` y ) ) = ( ( F ` A ) .+^ ( F ` y ) ) )
9	6, 8	eqeq12d 2220	. . . . 5 |- ( x = A -> ( ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) <-> ( F ` ( A .+ y ) ) = ( ( F ` A ) .+^ ( F ` y ) ) ) )
10	5, 9	imbi12d 234	. . . 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 ) ) ) ) )
11		eleq1 2268	. . . . . 6 |- ( y = B -> ( y e. X <-> B e. X ) )
12	11	3anbi3d 1331	. . . . 5 |- ( y = B -> ( ( ph /\ A e. X /\ y e. X ) <-> ( ph /\ A e. X /\ B e. X ) ) )
13		oveq2 5952	. . . . . . 7 |- ( y = B -> ( A .+ y ) = ( A .+ B ) )
14	13	fveq2d 5580	. . . . . 6 |- ( y = B -> ( F ` ( A .+ y ) ) = ( F ` ( A .+ B ) ) )
15		fveq2 5576	. . . . . . 7 |- ( y = B -> ( F ` y ) = ( F ` B ) )
16	15	oveq2d 5960	. . . . . 6 |- ( y = B -> ( ( F ` A ) .+^ ( F ` y ) ) = ( ( F ` A ) .+^ ( F ` B ) ) )
17	14, 16	eqeq12d 2220	. . . . 5 |- ( y = B -> ( ( F ` ( A .+ y ) ) = ( ( F ` A ) .+^ ( F ` y ) ) <-> ( F ` ( A .+ B ) ) = ( ( F ` A ) .+^ ( F ` B ) ) ) )
18	12, 17	imbi12d 234	. . . 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 ) ) ) ) )
19		ghmgrp.f	. . . 4 |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x .+ y ) ) = ( ( F ` x ) .+^ ( F ` y ) ) )
20	10, 18, 19	vtocl2g 2837	. . 3 |- ( ( A e. X /\ B e. X ) -> ( ( ph /\ A e. X /\ B e. X ) -> ( F ` ( A .+ B ) ) = ( ( F ` A ) .+^ ( F ` B ) ) ) )
21	2, 3, 20	syl2anc 411	. 2 |- ( ph -> ( ( ph /\ A e. X /\ B e. X ) -> ( F ` ( A .+ B ) ) = ( ( F ` A ) .+^ ( F ` B ) ) ) )
22	1, 2, 3, 21	mp3and 1353	1 |- ( ph -> ( F ` ( A .+ B ) ) = ( ( F ` A ) .+^ ( F ` B ) ) )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2176   ` cfv 5271  (class class class)co 5944

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  mhmid  13451  mhmmnd  13452  ghmgrp  13454  ghmcmn  13663