Theorem sgrp2rid2 17413

Description: A small semigroup (with two elements) with two right identities which are different if A =/= B. (Contributed by AV, 10-Feb-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
sgrp2rid2	|- ( ( A e. V /\ B e. W ) -> A. x e. S A. y e. S ( y .o. x ) = y )

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

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

Proof of Theorem sgrp2rid2

Step	Hyp	Ref	Expression
1		prid1g 4295	. . . 4 |- ( A e. V -> A e. { A , B } )
2		mgm2nsgrp.s	. . . 4 |- S = { A , B }
3	1, 2	syl6eleqr 2712	. . 3 |- ( A e. V -> A e. S )
4		prid2g 4296	. . . 4 |- ( B e. W -> B e. { A , B } )
5	4, 2	syl6eleqr 2712	. . 3 |- ( B e. W -> B e. S )
6		simpl 473	. . . . 5 |- ( ( A e. S /\ B e. S ) -> A e. S )
7		mgm2nsgrp.b	. . . . . 6 |- ( Base ` M ) = S
8		sgrp2nmnd.o	. . . . . 6 |- ( +g ` M ) = ( x e. S , y e. S |-> if ( x = A , A , B ) )
9		sgrp2nmnd.p	. . . . . 6 |- .o. = ( +g ` M )
10	2, 7, 8, 9	sgrp2nmndlem2 17411	. . . . 5 |- ( ( A e. S /\ A e. S ) -> ( A .o. A ) = A )
11	6, 10	syldan 487	. . . 4 |- ( ( A e. S /\ B e. S ) -> ( A .o. A ) = A )
12		oveq1 6657	. . . . . . 7 |- ( A = B -> ( A .o. A ) = ( B .o. A ) )
13		id 22	. . . . . . 7 |- ( A = B -> A = B )
14	12, 13	eqeq12d 2637	. . . . . 6 |- ( A = B -> ( ( A .o. A ) = A <-> ( B .o. A ) = B ) )
15	11, 14	syl5ib 234	. . . . 5 |- ( A = B -> ( ( A e. S /\ B e. S ) -> ( B .o. A ) = B ) )
16		simprl 794	. . . . . . 7 |- ( ( -. A = B /\ ( A e. S /\ B e. S ) ) -> A e. S )
17		simprr 796	. . . . . . 7 |- ( ( -. A = B /\ ( A e. S /\ B e. S ) ) -> B e. S )
18		df-ne 2795	. . . . . . . . 9 |- ( A =/= B <-> -. A = B )
19	18	biimpri 218	. . . . . . . 8 |- ( -. A = B -> A =/= B )
20	19	adantr 481	. . . . . . 7 |- ( ( -. A = B /\ ( A e. S /\ B e. S ) ) -> A =/= B )
21	2, 7, 8, 9	sgrp2nmndlem3 17412	. . . . . . 7 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> ( B .o. A ) = B )
22	16, 17, 20, 21	syl3anc 1326	. . . . . 6 |- ( ( -. A = B /\ ( A e. S /\ B e. S ) ) -> ( B .o. A ) = B )
23	22	ex 450	. . . . 5 |- ( -. A = B -> ( ( A e. S /\ B e. S ) -> ( B .o. A ) = B ) )
24	15, 23	pm2.61i 176	. . . 4 |- ( ( A e. S /\ B e. S ) -> ( B .o. A ) = B )
25	2, 7, 8, 9	sgrp2nmndlem2 17411	. . . . 5 |- ( ( A e. S /\ B e. S ) -> ( A .o. B ) = A )
26	13, 13	oveq12d 6668	. . . . . . . 8 |- ( A = B -> ( A .o. A ) = ( B .o. B ) )
27	26, 13	eqeq12d 2637	. . . . . . 7 |- ( A = B -> ( ( A .o. A ) = A <-> ( B .o. B ) = B ) )
28	11, 27	syl5ib 234	. . . . . 6 |- ( A = B -> ( ( A e. S /\ B e. S ) -> ( B .o. B ) = B ) )
29	2, 7, 8, 9	sgrp2nmndlem3 17412	. . . . . . . 8 |- ( ( B e. S /\ B e. S /\ A =/= B ) -> ( B .o. B ) = B )
30	17, 17, 20, 29	syl3anc 1326	. . . . . . 7 |- ( ( -. A = B /\ ( A e. S /\ B e. S ) ) -> ( B .o. B ) = B )
31	30	ex 450	. . . . . 6 |- ( -. A = B -> ( ( A e. S /\ B e. S ) -> ( B .o. B ) = B ) )
32	28, 31	pm2.61i 176	. . . . 5 |- ( ( A e. S /\ B e. S ) -> ( B .o. B ) = B )
33	25, 32	jca 554	. . . 4 |- ( ( A e. S /\ B e. S ) -> ( ( A .o. B ) = A /\ ( B .o. B ) = B ) )
34	11, 24, 33	jca31 557	. . 3 |- ( ( A e. S /\ B e. S ) -> ( ( ( A .o. A ) = A /\ ( B .o. A ) = B ) /\ ( ( A .o. B ) = A /\ ( B .o. B ) = B ) ) )
35	3, 5, 34	syl2an 494	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( ( A .o. A ) = A /\ ( B .o. A ) = B ) /\ ( ( A .o. B ) = A /\ ( B .o. B ) = B ) ) )
36	2	raleqi 3142	. . . . 5 |- ( A. y e. S ( y .o. x ) = y <-> A. y e. { A , B } ( y .o. x ) = y )
37		oveq1 6657	. . . . . . 7 |- ( y = A -> ( y .o. x ) = ( A .o. x ) )
38		id 22	. . . . . . 7 |- ( y = A -> y = A )
39	37, 38	eqeq12d 2637	. . . . . 6 |- ( y = A -> ( ( y .o. x ) = y <-> ( A .o. x ) = A ) )
40		oveq1 6657	. . . . . . 7 |- ( y = B -> ( y .o. x ) = ( B .o. x ) )
41		id 22	. . . . . . 7 |- ( y = B -> y = B )
42	40, 41	eqeq12d 2637	. . . . . 6 |- ( y = B -> ( ( y .o. x ) = y <-> ( B .o. x ) = B ) )
43	39, 42	ralprg 4234	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( A. y e. { A , B } ( y .o. x ) = y <-> ( ( A .o. x ) = A /\ ( B .o. x ) = B ) ) )
44	36, 43	syl5bb 272	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( A. y e. S ( y .o. x ) = y <-> ( ( A .o. x ) = A /\ ( B .o. x ) = B ) ) )
45	44	ralbidv 2986	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A. x e. S A. y e. S ( y .o. x ) = y <-> A. x e. S ( ( A .o. x ) = A /\ ( B .o. x ) = B ) ) )
46	2	raleqi 3142	. . . 4 |- ( A. x e. S ( ( A .o. x ) = A /\ ( B .o. x ) = B ) <-> A. x e. { A , B } ( ( A .o. x ) = A /\ ( B .o. x ) = B ) )
47		oveq2 6658	. . . . . . 7 |- ( x = A -> ( A .o. x ) = ( A .o. A ) )
48	47	eqeq1d 2624	. . . . . 6 |- ( x = A -> ( ( A .o. x ) = A <-> ( A .o. A ) = A ) )
49		oveq2 6658	. . . . . . 7 |- ( x = A -> ( B .o. x ) = ( B .o. A ) )
50	49	eqeq1d 2624	. . . . . 6 |- ( x = A -> ( ( B .o. x ) = B <-> ( B .o. A ) = B ) )
51	48, 50	anbi12d 747	. . . . 5 |- ( x = A -> ( ( ( A .o. x ) = A /\ ( B .o. x ) = B ) <-> ( ( A .o. A ) = A /\ ( B .o. A ) = B ) ) )
52		oveq2 6658	. . . . . . 7 |- ( x = B -> ( A .o. x ) = ( A .o. B ) )
53	52	eqeq1d 2624	. . . . . 6 |- ( x = B -> ( ( A .o. x ) = A <-> ( A .o. B ) = A ) )
54		oveq2 6658	. . . . . . 7 |- ( x = B -> ( B .o. x ) = ( B .o. B ) )
55	54	eqeq1d 2624	. . . . . 6 |- ( x = B -> ( ( B .o. x ) = B <-> ( B .o. B ) = B ) )
56	53, 55	anbi12d 747	. . . . 5 |- ( x = B -> ( ( ( A .o. x ) = A /\ ( B .o. x ) = B ) <-> ( ( A .o. B ) = A /\ ( B .o. B ) = B ) ) )
57	51, 56	ralprg 4234	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } ( ( A .o. x ) = A /\ ( B .o. x ) = B ) <-> ( ( ( A .o. A ) = A /\ ( B .o. A ) = B ) /\ ( ( A .o. B ) = A /\ ( B .o. B ) = B ) ) ) )
58	46, 57	syl5bb 272	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A. x e. S ( ( A .o. x ) = A /\ ( B .o. x ) = B ) <-> ( ( ( A .o. A ) = A /\ ( B .o. A ) = B ) /\ ( ( A .o. B ) = A /\ ( B .o. B ) = B ) ) ) )
59	45, 58	bitrd 268	. 2 |- ( ( A e. V /\ B e. W ) -> ( A. x e. S A. y e. S ( y .o. x ) = y <-> ( ( ( A .o. A ) = A /\ ( B .o. A ) = B ) /\ ( ( A .o. B ) = A /\ ( B .o. B ) = B ) ) ) )
60	35, 59	mpbird 247	1 |- ( ( A e. V /\ B e. W ) -> A. x e. S A. y e. S ( y .o. x ) = y )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   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:  sgrp2rid2ex  17414