Theorem mgm2nsgrplem3 17407

Description: Lemma 3 for mgm2nsgrp 17409. (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 4295	. . 3 |- ( A e. V -> A e. { A , B } )
2		mgm2nsgrp.s	. . 3 |- S = { A , B }
3	1, 2	syl6eleqr 2712	. 2 |- ( A e. V -> A e. S )
4		prid2g 4296	. . 3 |- ( B e. W -> B e. { A , B } )
5	4, 2	syl6eleqr 2712	. 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 2644	. . . 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 794	. . . . 5 |- ( ( ( A e. S /\ B e. S ) /\ ( x = A /\ y = ( A .o. B ) ) ) -> x = A )
11		simpr 477	. . . . . 6 |- ( ( x = A /\ y = ( A .o. B ) ) -> y = ( A .o. B ) )
12		ifeq1 4090	. . . . . . . . . . 11 |- ( B = A -> if ( ( x = A /\ y = A ) , B , A ) = if ( ( x = A /\ y = A ) , A , A ) )
13		ifid 4125	. . . . . . . . . . 11 |- if ( ( x = A /\ y = A ) , A , A ) = A
14	12, 13	syl6eq 2672	. . . . . . . . . 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 2626	. . . . . . . . . . . . . . . . 17 |- ( y = B -> ( y = A <-> B = A ) )
17	16	biimpcd 239	. . . . . . . . . . . . . . . 16 |- ( y = A -> ( y = B -> B = A ) )
18	17	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( x = A /\ y = A ) -> ( y = B -> B = A ) )
19	18	com12 32	. . . . . . . . . . . . . 14 |- ( y = B -> ( ( x = A /\ y = A ) -> B = A ) )
20	19	adantl 482	. . . . . . . . . . . . 13 |- ( ( x = A /\ y = B ) -> ( ( x = A /\ y = A ) -> B = A ) )
21	20	con3d 148	. . . . . . . . . . . 12 |- ( ( x = A /\ y = B ) -> ( -. B = A -> -. ( x = A /\ y = A ) ) )
22	21	impcom 446	. . . . . . . . . . 11 |- ( ( -. B = A /\ ( x = A /\ y = B ) ) -> -. ( x = A /\ y = A ) )
23	22	iffalsed 4097	. . . . . . . . . 10 |- ( ( -. B = A /\ ( x = A /\ y = B ) ) -> if ( ( x = A /\ y = A ) , B , A ) = A )
24	23	ex 450	. . . . . . . . 9 |- ( -. B = A -> ( ( x = A /\ y = B ) -> if ( ( x = A /\ y = A ) , B , A ) = A ) )
25	15, 24	pm2.61i 176	. . . . . . . 8 |- ( ( x = A /\ y = B ) -> if ( ( x = A /\ y = A ) , B , A ) = A )
26	25	adantl 482	. . . . . . 7 |- ( ( ( A e. S /\ B e. S ) /\ ( x = A /\ y = B ) ) -> if ( ( x = A /\ y = A ) , B , A ) = A )
27		simpl 473	. . . . . . 7 |- ( ( A e. S /\ B e. S ) -> A e. S )
28		simpr 477	. . . . . . 7 |- ( ( A e. S /\ B e. S ) -> B e. S )
29	9, 26, 27, 28, 27	ovmpt2d 6788	. . . . . 6 |- ( ( A e. S /\ B e. S ) -> ( A .o. B ) = A )
30	11, 29	sylan9eqr 2678	. . . . 5 |- ( ( ( A e. S /\ B e. S ) /\ ( x = A /\ y = ( A .o. B ) ) ) -> y = A )
31	10, 30	jca 554	. . . 4 |- ( ( ( A e. S /\ B e. S ) /\ ( x = A /\ y = ( A .o. B ) ) ) -> ( x = A /\ y = A ) )
32	31	iftrued 4094	. . 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 2701	. . 3 |- ( ( A e. S /\ B e. S ) -> ( A .o. B ) e. S )
34	9, 32, 27, 33, 28	ovmpt2d 6788	. 2 |- ( ( A e. S /\ B e. S ) -> ( A .o. ( A .o. B ) ) = B )
35	3, 5, 34	syl2an 494	1 |- ( ( A e. V /\ B e. W ) -> ( A .o. ( A .o. B ) ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   {cpr 4179   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   +g cplusg 15941

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  mgm2nsgrplem4  17408