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Theorem mgm2nsgrplem3 16654
Description: Lemma 3 for mgm2nsgrp 16656. (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 4078 . . 3  |-  ( A  e.  V  ->  A  e.  { A ,  B } )
2 mgm2nsgrp.s . . 3  |-  S  =  { A ,  B }
31, 2syl6eleqr 2540 . 2  |-  ( A  e.  V  ->  A  e.  S )
4 prid2g 4079 . . 3  |-  ( B  e.  W  ->  B  e.  { A ,  B } )
54, 2syl6eleqr 2540 . 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 2473 . . . 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 764 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  A  /\  y  =  ( A  .o.  B ) ) )  ->  x  =  A )
11 simpr 463 . . . . . 6  |-  ( ( x  =  A  /\  y  =  ( A  .o.  B ) )  -> 
y  =  ( A  .o.  B ) )
12 ifeq1 3885 . . . . . . . . . . 11  |-  ( B  =  A  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  if ( ( x  =  A  /\  y  =  A ) ,  A ,  A ) )
13 ifid 3918 . . . . . . . . . . 11  |-  if ( ( x  =  A  /\  y  =  A ) ,  A ,  A )  =  A
1412, 13syl6eq 2501 . . . . . . . . . 10  |-  ( B  =  A  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
1514a1d 26 . . . . . . . . 9  |-  ( B  =  A  ->  (
( x  =  A  /\  y  =  B )  ->  if (
( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A ) )
16 eqeq1 2455 . . . . . . . . . . . . . . . . 17  |-  ( y  =  B  ->  (
y  =  A  <->  B  =  A ) )
1716biimpcd 228 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  (
y  =  B  ->  B  =  A )
)
1817adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( x  =  A  /\  y  =  A )  ->  ( y  =  B  ->  B  =  A ) )
1918com12 32 . . . . . . . . . . . . . 14  |-  ( y  =  B  ->  (
( x  =  A  /\  y  =  A )  ->  B  =  A ) )
2019adantl 468 . . . . . . . . . . . . 13  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  =  A  /\  y  =  A )  ->  B  =  A ) )
2120con3d 139 . . . . . . . . . . . 12  |-  ( ( x  =  A  /\  y  =  B )  ->  ( -.  B  =  A  ->  -.  (
x  =  A  /\  y  =  A )
) )
2221impcom 432 . . . . . . . . . . 11  |-  ( ( -.  B  =  A  /\  ( x  =  A  /\  y  =  B ) )  ->  -.  ( x  =  A  /\  y  =  A ) )
2322iffalsed 3892 . . . . . . . . . 10  |-  ( ( -.  B  =  A  /\  ( x  =  A  /\  y  =  B ) )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
2423ex 436 . . . . . . . . 9  |-  ( -.  B  =  A  -> 
( ( x  =  A  /\  y  =  B )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A ) )
2515, 24pm2.61i 168 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
2625adantl 468 . . . . . . 7  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  A  /\  y  =  B ) )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
27 simpl 459 . . . . . . 7  |-  ( ( A  e.  S  /\  B  e.  S )  ->  A  e.  S )
28 simpr 463 . . . . . . 7  |-  ( ( A  e.  S  /\  B  e.  S )  ->  B  e.  S )
299, 26, 27, 28, 27ovmpt2d 6424 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  B
)  =  A )
3011, 29sylan9eqr 2507 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  A  /\  y  =  ( A  .o.  B ) ) )  ->  y  =  A )
3110, 30jca 535 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  A  /\  y  =  ( A  .o.  B ) ) )  ->  ( x  =  A  /\  y  =  A ) )
3231iftrued 3889 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  A  /\  y  =  ( A  .o.  B ) ) )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  B )
3329, 27eqeltrd 2529 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  B
)  e.  S )
349, 32, 27, 33, 28ovmpt2d 6424 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  ( A  .o.  B ) )  =  B )
353, 5, 34syl2an 480 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 371    = wceq 1444    e. wcel 1887   ifcif 3881   {cpr 3970   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   Basecbs 15121   +g cplusg 15190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295
This theorem is referenced by:  mgm2nsgrplem4  16655
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