Theorem mgm2nsgrplem3 16165

Description: Lemma 3 for mgm2nsgrp 16167. (Contributed by AV, 28-Jan-2020.)

Hypotheses

Ref	Expression
mgm2nsgrp.s	|- S = { A , B }
mgm2nsgrp.b	|- ( Base ` M ) = S
mgm2nsgrp.o	|- ( +g ` M ) = ( x e. S , y e. S |-> if ( ( x = A /\ y = A ) , B , A ) )
mgm2nsgrp.p	|- .o. = ( +g ` M )

Assertion

Ref	Expression
mgm2nsgrplem3	|- ( ( A e. V /\ B e. W ) -> ( A .o. ( A .o. B ) ) = B )

Distinct variable groups:   x, S, y   x, A, y   x, B, y   x, M   x, .o. , y

Allowed substitution hints:   M( y)   V( x, y)   W( x, y)

Proof of Theorem mgm2nsgrplem3

Step	Hyp	Ref	Expression
1		prid1g 4138	. . 3 |- ( A e. V -> A e. { A , B } )
2		mgm2nsgrp.s	. . 3 |- S = { A , B }
3	1, 2	syl6eleqr 2556	. 2 |- ( A e. V -> A e. S )
4		prid2g 4139	. . 3 |- ( B e. W -> B e. { A , B } )
5	4, 2	syl6eleqr 2556	. 2 |- ( B e. W -> B e. S )
6		mgm2nsgrp.p	. . . . 5 |- .o. = ( +g ` M )
7		mgm2nsgrp.o	. . . . 5 |- ( +g ` M ) = ( x e. S , y e. S |-> if ( ( x = A /\ y = A ) , B , A ) )
8	6, 7	eqtri 2486	. . . 4 |- .o. = ( x e. S , y e. S |-> if ( ( x = A /\ y = A ) , B , A ) )
9	8	a1i 11	. . 3 |- ( ( A e. S /\ B e. S ) -> .o. = ( x e. S , y e. S |-> if ( ( x = A /\ y = A ) , B , A ) ) )
10		simprl 756	. . . . 5 |- ( ( ( A e. S /\ B e. S ) /\ ( x = A /\ y = ( A .o. B ) ) ) -> x = A )
11		simpr 461	. . . . . 6 |- ( ( x = A /\ y = ( A .o. B ) ) -> y = ( A .o. B ) )
12		ifeq1 3948	. . . . . . . . . . 11 |- ( B = A -> if ( ( x = A /\ y = A ) , B , A ) = if ( ( x = A /\ y = A ) , A , A ) )
13		ifid 3981	. . . . . . . . . . 11 |- if ( ( x = A /\ y = A ) , A , A ) = A
14	12, 13	syl6eq 2514	. . . . . . . . . 10 |- ( B = A -> if ( ( x = A /\ y = A ) , B , A ) = A )
15	14	a1d 25	. . . . . . . . 9 |- ( B = A -> ( ( x = A /\ y = B ) -> if ( ( x = A /\ y = A ) , B , A ) = A ) )
16		eqeq1 2461	. . . . . . . . . . . . . . . . 17 |- ( y = B -> ( y = A <-> B = A ) )
17	16	biimpcd 224	. . . . . . . . . . . . . . . 16 |- ( y = A -> ( y = B -> B = A ) )
18	17	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( x = A /\ y = A ) -> ( y = B -> B = A ) )
19	18	com12 31	. . . . . . . . . . . . . 14 |- ( y = B -> ( ( x = A /\ y = A ) -> B = A ) )
20	19	adantl 466	. . . . . . . . . . . . 13 |- ( ( x = A /\ y = B ) -> ( ( x = A /\ y = A ) -> B = A ) )
21	20	con3d 133	. . . . . . . . . . . 12 |- ( ( x = A /\ y = B ) -> ( -. B = A -> -. ( x = A /\ y = A ) ) )
22	21	impcom 430	. . . . . . . . . . 11 |- ( ( -. B = A /\ ( x = A /\ y = B ) ) -> -. ( x = A /\ y = A ) )
23	22	iffalsed 3955	. . . . . . . . . 10 |- ( ( -. B = A /\ ( x = A /\ y = B ) ) -> if ( ( x = A /\ y = A ) , B , A ) = A )
24	23	ex 434	. . . . . . . . 9 |- ( -. B = A -> ( ( x = A /\ y = B ) -> if ( ( x = A /\ y = A ) , B , A ) = A ) )
25	15, 24	pm2.61i 164	. . . . . . . 8 |- ( ( x = A /\ y = B ) -> if ( ( x = A /\ y = A ) , B , A ) = A )
26	25	adantl 466	. . . . . . 7 |- ( ( ( A e. S /\ B e. S ) /\ ( x = A /\ y = B ) ) -> if ( ( x = A /\ y = A ) , B , A ) = A )
27		simpl 457	. . . . . . 7 |- ( ( A e. S /\ B e. S ) -> A e. S )
28		simpr 461	. . . . . . 7 |- ( ( A e. S /\ B e. S ) -> B e. S )
29	9, 26, 27, 28, 27	ovmpt2d 6429	. . . . . 6 |- ( ( A e. S /\ B e. S ) -> ( A .o. B ) = A )
30	11, 29	sylan9eqr 2520	. . . . 5 |- ( ( ( A e. S /\ B e. S ) /\ ( x = A /\ y = ( A .o. B ) ) ) -> y = A )
31	10, 30	jca 532	. . . 4 |- ( ( ( A e. S /\ B e. S ) /\ ( x = A /\ y = ( A .o. B ) ) ) -> ( x = A /\ y = A ) )
32	31	iftrued 3952	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( x = A /\ y = ( A .o. B ) ) ) -> if ( ( x = A /\ y = A ) , B , A ) = B )
33	29, 27	eqeltrd 2545	. . 3 |- ( ( A e. S /\ B e. S ) -> ( A .o. B ) e. S )
34	9, 32, 27, 33, 28	ovmpt2d 6429	. 2 |- ( ( A e. S /\ B e. S ) -> ( A .o. ( A .o. B ) ) = B )
35	3, 5, 34	syl2an 477	1 |- ( ( A e. V /\ B e. W ) -> ( A .o. ( A .o. B ) ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1395   e. wcel 1819   ifcif 3944   {cpr 4034   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   Basecbs 14644   +g cplusg 14712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301

This theorem is referenced by:  mgm2nsgrplem4  16166