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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
StepHypRef Expression
1 prid1g 4138 . . 3  |-  ( A  e.  V  ->  A  e.  { A ,  B } )
2 mgm2nsgrp.s . . 3  |-  S  =  { A ,  B }
31, 2syl6eleqr 2556 . 2  |-  ( A  e.  V  ->  A  e.  S )
4 prid2g 4139 . . 3  |-  ( B  e.  W  ->  B  e.  { A ,  B } )
54, 2syl6eleqr 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 ) )
86, 7eqtri 2486 . . . 4  |-  .o.  =  ( x  e.  S ,  y  e.  S  |->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A ) )
98a1i 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
1412, 13syl6eq 2514 . . . . . . . . . 10  |-  ( B  =  A  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
1514a1d 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 ) )
1716biimpcd 224 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  (
y  =  B  ->  B  =  A )
)
1817adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( x  =  A  /\  y  =  A )  ->  ( y  =  B  ->  B  =  A ) )
1918com12 31 . . . . . . . . . . . . . 14  |-  ( y  =  B  ->  (
( x  =  A  /\  y  =  A )  ->  B  =  A ) )
2019adantl 466 . . . . . . . . . . . . 13  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  =  A  /\  y  =  A )  ->  B  =  A ) )
2120con3d 133 . . . . . . . . . . . 12  |-  ( ( x  =  A  /\  y  =  B )  ->  ( -.  B  =  A  ->  -.  (
x  =  A  /\  y  =  A )
) )
2221impcom 430 . . . . . . . . . . 11  |-  ( ( -.  B  =  A  /\  ( x  =  A  /\  y  =  B ) )  ->  -.  ( x  =  A  /\  y  =  A ) )
2322iffalsed 3955 . . . . . . . . . 10  |-  ( ( -.  B  =  A  /\  ( x  =  A  /\  y  =  B ) )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
2423ex 434 . . . . . . . . 9  |-  ( -.  B  =  A  -> 
( ( x  =  A  /\  y  =  B )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A ) )
2515, 24pm2.61i 164 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
2625adantl 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 )
299, 26, 27, 28, 27ovmpt2d 6429 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  B
)  =  A )
3011, 29sylan9eqr 2520 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  A  /\  y  =  ( A  .o.  B ) ) )  ->  y  =  A )
3110, 30jca 532 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  A  /\  y  =  ( A  .o.  B ) ) )  ->  ( x  =  A  /\  y  =  A ) )
3231iftrued 3952 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  A  /\  y  =  ( A  .o.  B ) ) )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  B )
3329, 27eqeltrd 2545 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  B
)  e.  S )
349, 32, 27, 33, 28ovmpt2d 6429 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  ( A  .o.  B ) )  =  B )
353, 5, 34syl2an 477 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  .o.  ( A  .o.  B ) )  =  B )
Colors of variables: wff setvar class
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
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