Theorem sgrp2nmndlem3 17412

Description: Lemma 3 for sgrp2nmnd 17417. (Contributed by AV, 29-Jan-2020.)

Hypotheses

Ref	Expression
mgm2nsgrp.s	|- S = { A , B }
mgm2nsgrp.b	|- ( Base ` M ) = S
sgrp2nmnd.o	|- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )
sgrp2nmnd.p	|- .o. = ( +g ` M )

Assertion

Ref	Expression
sgrp2nmndlem3	|- ( ( C e. S /\ B e. S /\ A =/= B ) -> ( B .o. C ) = B )

Distinct variable groups:   x, S, y   x, A, y   x, B, y   x, M   x, C, y

Allowed substitution hints:   M( y)   .o. ( x, y)

Proof of Theorem sgrp2nmndlem3

Step	Hyp	Ref	Expression
1		sgrp2nmnd.p	. . . 4 |- .o. = ( +g ` M )
2		sgrp2nmnd.o	. . . 4 |- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )
3	1, 2	eqtri 2644	. . 3 |- .o. = ( x e. S , y e. S |-> if ( x = A , A , B ) )
4	3	a1i 11	. 2 |- ( ( C e. S /\ B e. S /\ A =/= B ) -> .o. = ( x e. S , y e. S |-> if ( x = A , A , B ) ) )
5		df-ne 2795	. . . . . 6 |- ( A =/= B <-> -. A = B )
6		eqcom 2629	. . . . . . . . 9 |- ( A = x <-> x = A )
7		eqeq2 2633	. . . . . . . . . 10 |- ( x = B -> ( A = x <-> A = B ) )
8	7	adantr 481	. . . . . . . . 9 |- ( ( x = B /\ y = C ) -> ( A = x <-> A = B ) )
9	6, 8	syl5rbbr 275	. . . . . . . 8 |- ( ( x = B /\ y = C ) -> ( A = B <-> x = A ) )
10	9	notbid 308	. . . . . . 7 |- ( ( x = B /\ y = C ) -> ( -. A = B <-> -. x = A ) )
11	10	biimpcd 239	. . . . . 6 |- ( -. A = B -> ( ( x = B /\ y = C ) -> -. x = A ) )
12	5, 11	sylbi 207	. . . . 5 |- ( A =/= B -> ( ( x = B /\ y = C ) -> -. x = A ) )
13	12	3ad2ant3 1084	. . . 4 |- ( ( C e. S /\ B e. S /\ A =/= B ) -> ( ( x = B /\ y = C ) -> -. x = A ) )
14	13	imp 445	. . 3 |- ( ( ( C e. S /\ B e. S /\ A =/= B ) /\ ( x = B /\ y = C ) ) -> -. x = A )
15	14	iffalsed 4097	. 2 |- ( ( ( C e. S /\ B e. S /\ A =/= B ) /\ ( x = B /\ y = C ) ) -> if ( x = A , A , B ) = B )
16		simp2 1062	. 2 |- ( ( C e. S /\ B e. S /\ A =/= B ) -> B e. S )
17		simp1 1061	. 2 |- ( ( C e. S /\ B e. S /\ A =/= B ) -> C e. S )
18	4, 15, 16, 17, 16	ovmpt2d 6788	1 |- ( ( C e. S /\ B e. S /\ A =/= B ) -> ( B .o. C ) = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   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-ne 2795  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:  sgrp2rid2  17413  sgrp2nmndlem4  17415  sgrp2nmndlem5  17416