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Theorem sgrp2rid2 16023
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
StepHypRef Expression
1 prid1g 4121 . . . 4  |-  ( A  e.  V  ->  A  e.  { A ,  B } )
2 mgm2nsgrp.s . . . 4  |-  S  =  { A ,  B }
31, 2syl6eleqr 2542 . . 3  |-  ( A  e.  V  ->  A  e.  S )
4 prid2g 4122 . . . 4  |-  ( B  e.  W  ->  B  e.  { A ,  B } )
54, 2syl6eleqr 2542 . . 3  |-  ( B  e.  W  ->  B  e.  S )
6 simpl 457 . . . . 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 )
102, 7, 8, 9sgrp2nmndlem2 16021 . . . . 5  |-  ( ( A  e.  S  /\  A  e.  S )  ->  ( A  .o.  A
)  =  A )
116, 10syldan 470 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  A
)  =  A )
12 oveq1 6288 . . . . . . 7  |-  ( A  =  B  ->  ( A  .o.  A )  =  ( B  .o.  A
) )
13 id 22 . . . . . . 7  |-  ( A  =  B  ->  A  =  B )
1412, 13eqeq12d 2465 . . . . . 6  |-  ( A  =  B  ->  (
( A  .o.  A
)  =  A  <->  ( B  .o.  A )  =  B ) )
1511, 14syl5ib 219 . . . . 5  |-  ( A  =  B  ->  (
( A  e.  S  /\  B  e.  S
)  ->  ( B  .o.  A )  =  B ) )
16 simprl 756 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( A  e.  S  /\  B  e.  S ) )  ->  A  e.  S )
17 simprr 757 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( A  e.  S  /\  B  e.  S ) )  ->  B  e.  S )
18 df-ne 2640 . . . . . . . . 9  |-  ( A  =/=  B  <->  -.  A  =  B )
1918biimpri 206 . . . . . . . 8  |-  ( -.  A  =  B  ->  A  =/=  B )
2019adantr 465 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( A  e.  S  /\  B  e.  S ) )  ->  A  =/=  B )
212, 7, 8, 9sgrp2nmndlem3 16022 . . . . . . 7  |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  -> 
( B  .o.  A
)  =  B )
2216, 17, 20, 21syl3anc 1229 . . . . . 6  |-  ( ( -.  A  =  B  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( B  .o.  A
)  =  B )
2322ex 434 . . . . 5  |-  ( -.  A  =  B  -> 
( ( A  e.  S  /\  B  e.  S )  ->  ( B  .o.  A )  =  B ) )
2415, 23pm2.61i 164 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( B  .o.  A
)  =  B )
252, 7, 8, 9sgrp2nmndlem2 16021 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  B
)  =  A )
2613, 13oveq12d 6299 . . . . . . . 8  |-  ( A  =  B  ->  ( A  .o.  A )  =  ( B  .o.  B
) )
2726, 13eqeq12d 2465 . . . . . . 7  |-  ( A  =  B  ->  (
( A  .o.  A
)  =  A  <->  ( B  .o.  B )  =  B ) )
2811, 27syl5ib 219 . . . . . 6  |-  ( A  =  B  ->  (
( A  e.  S  /\  B  e.  S
)  ->  ( B  .o.  B )  =  B ) )
292, 7, 8, 9sgrp2nmndlem3 16022 . . . . . . . 8  |-  ( ( B  e.  S  /\  B  e.  S  /\  A  =/=  B )  -> 
( B  .o.  B
)  =  B )
3017, 17, 20, 29syl3anc 1229 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( B  .o.  B
)  =  B )
3130ex 434 . . . . . 6  |-  ( -.  A  =  B  -> 
( ( A  e.  S  /\  B  e.  S )  ->  ( B  .o.  B )  =  B ) )
3228, 31pm2.61i 164 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( B  .o.  B
)  =  B )
3325, 32jca 532 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( ( A  .o.  B )  =  A  /\  ( B  .o.  B )  =  B ) )
3411, 24, 33jca31 534 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( ( ( A  .o.  A )  =  A  /\  ( B  .o.  A )  =  B )  /\  (
( A  .o.  B
)  =  A  /\  ( B  .o.  B )  =  B ) ) )
353, 5, 34syl2an 477 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( ( A  .o.  A )  =  A  /\  ( B  .o.  A )  =  B )  /\  (
( A  .o.  B
)  =  A  /\  ( B  .o.  B )  =  B ) ) )
362raleqi 3044 . . . . 5  |-  ( A. y  e.  S  (
y  .o.  x )  =  y  <->  A. y  e.  { A ,  B } 
( y  .o.  x
)  =  y )
37 oveq1 6288 . . . . . . 7  |-  ( y  =  A  ->  (
y  .o.  x )  =  ( A  .o.  x ) )
38 id 22 . . . . . . 7  |-  ( y  =  A  ->  y  =  A )
3937, 38eqeq12d 2465 . . . . . 6  |-  ( y  =  A  ->  (
( y  .o.  x
)  =  y  <->  ( A  .o.  x )  =  A ) )
40 oveq1 6288 . . . . . . 7  |-  ( y  =  B  ->  (
y  .o.  x )  =  ( B  .o.  x ) )
41 id 22 . . . . . . 7  |-  ( y  =  B  ->  y  =  B )
4240, 41eqeq12d 2465 . . . . . 6  |-  ( y  =  B  ->  (
( y  .o.  x
)  =  y  <->  ( B  .o.  x )  =  B ) )
4339, 42ralprg 4063 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. y  e. 
{ A ,  B }  ( y  .o.  x )  =  y  <-> 
( ( A  .o.  x )  =  A  /\  ( B  .o.  x )  =  B ) ) )
4436, 43syl5bb 257 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. y  e.  S  ( y  .o.  x )  =  y  <-> 
( ( A  .o.  x )  =  A  /\  ( B  .o.  x )  =  B ) ) )
4544ralbidv 2882 . . 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 ) ) )
462raleqi 3044 . . . 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 6289 . . . . . . 7  |-  ( x  =  A  ->  ( A  .o.  x )  =  ( A  .o.  A
) )
4847eqeq1d 2445 . . . . . 6  |-  ( x  =  A  ->  (
( A  .o.  x
)  =  A  <->  ( A  .o.  A )  =  A ) )
49 oveq2 6289 . . . . . . 7  |-  ( x  =  A  ->  ( B  .o.  x )  =  ( B  .o.  A
) )
5049eqeq1d 2445 . . . . . 6  |-  ( x  =  A  ->  (
( B  .o.  x
)  =  B  <->  ( B  .o.  A )  =  B ) )
5148, 50anbi12d 710 . . . . 5  |-  ( x  =  A  ->  (
( ( A  .o.  x )  =  A  /\  ( B  .o.  x )  =  B )  <->  ( ( A  .o.  A )  =  A  /\  ( B  .o.  A )  =  B ) ) )
52 oveq2 6289 . . . . . . 7  |-  ( x  =  B  ->  ( A  .o.  x )  =  ( A  .o.  B
) )
5352eqeq1d 2445 . . . . . 6  |-  ( x  =  B  ->  (
( A  .o.  x
)  =  A  <->  ( A  .o.  B )  =  A ) )
54 oveq2 6289 . . . . . . 7  |-  ( x  =  B  ->  ( B  .o.  x )  =  ( B  .o.  B
) )
5554eqeq1d 2445 . . . . . 6  |-  ( x  =  B  ->  (
( B  .o.  x
)  =  B  <->  ( B  .o.  B )  =  B ) )
5653, 55anbi12d 710 . . . . 5  |-  ( x  =  B  ->  (
( ( A  .o.  x )  =  A  /\  ( B  .o.  x )  =  B )  <->  ( ( A  .o.  B )  =  A  /\  ( B  .o.  B )  =  B ) ) )
5751, 56ralprg 4063 . . . 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 ) ) ) )
5846, 57syl5bb 257 . . 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 ) ) ) )
5945, 58bitrd 253 . 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 ) ) ) )
6035, 59mpbird 232 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A. x  e.  S  A. y  e.  S  ( y  .o.  x
)  =  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   ifcif 3926   {cpr 4016   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   Basecbs 14614   +g cplusg 14679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286
This theorem is referenced by:  sgrp2rid2ex  16024
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