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Theorem mgm2nsgrplem2 16016
Description: Lemma 2 for mgm2nsgrp 16019. (Contributed by AV, 27-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
mgm2nsgrplem2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  .o.  A )  .o.  B
)  =  A )
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 mgm2nsgrplem2
StepHypRef Expression
1 prid1g 4121 . . 3  |-  ( A  e.  V  ->  A  e.  { A ,  B } )
2 mgm2nsgrp.s . . 3  |-  S  =  { A ,  B }
31, 2syl6eleqr 2542 . 2  |-  ( A  e.  V  ->  A  e.  S )
4 prid2g 4122 . . 3  |-  ( B  e.  W  ->  B  e.  { A ,  B } )
54, 2syl6eleqr 2542 . 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 2472 . . . 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 ifeq1 3930 . . . . . . 7  |-  ( B  =  A  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  if ( ( x  =  A  /\  y  =  A ) ,  A ,  A ) )
11 ifid 3963 . . . . . . 7  |-  if ( ( x  =  A  /\  y  =  A ) ,  A ,  A )  =  A
1210, 11syl6eq 2500 . . . . . 6  |-  ( B  =  A  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
1312a1d 25 . . . . 5  |-  ( B  =  A  ->  (
y  =  B  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A ) )
14 eqeq1 2447 . . . . . . . . . . 11  |-  ( y  =  B  ->  (
y  =  A  <->  B  =  A ) )
1514bicomd 201 . . . . . . . . . 10  |-  ( y  =  B  ->  ( B  =  A  <->  y  =  A ) )
1615notbid 294 . . . . . . . . 9  |-  ( y  =  B  ->  ( -.  B  =  A  <->  -.  y  =  A ) )
1716biimpac 486 . . . . . . . 8  |-  ( ( -.  B  =  A  /\  y  =  B )  ->  -.  y  =  A )
1817intnand 916 . . . . . . 7  |-  ( ( -.  B  =  A  /\  y  =  B )  ->  -.  (
x  =  A  /\  y  =  A )
)
1918iffalsed 3937 . . . . . 6  |-  ( ( -.  B  =  A  /\  y  =  B )  ->  if (
( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
2019ex 434 . . . . 5  |-  ( -.  B  =  A  -> 
( y  =  B  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A ) )
2113, 20pm2.61i 164 . . . 4  |-  ( y  =  B  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
2221ad2antll 728 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  ( A  .o.  A )  /\  y  =  B ) )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
23 iftrue 3932 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  B )
2423adantl 466 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  A  /\  y  =  A ) )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  B )
25 simpl 457 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S )  ->  A  e.  S )
26 simpr 461 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S )  ->  B  e.  S )
279, 24, 25, 25, 26ovmpt2d 6415 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  A
)  =  B )
2827, 26eqeltrd 2531 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  A
)  e.  S )
299, 22, 28, 26, 25ovmpt2d 6415 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( ( A  .o.  A )  .o.  B
)  =  A )
303, 5, 29syl2an 477 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  .o.  A )  .o.  B
)  =  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   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:  mgm2nsgrplem4  16018
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