MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mgm2nsgrplem2 Structured version   Unicode version

Theorem mgm2nsgrplem2 16251
Description: Lemma 2 for mgm2nsgrp 16254. (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 4075 . . 3  |-  ( A  e.  V  ->  A  e.  { A ,  B } )
2 mgm2nsgrp.s . . 3  |-  S  =  { A ,  B }
31, 2syl6eleqr 2499 . 2  |-  ( A  e.  V  ->  A  e.  S )
4 prid2g 4076 . . 3  |-  ( B  e.  W  ->  B  e.  { A ,  B } )
54, 2syl6eleqr 2499 . 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 2429 . . . 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 3886 . . . . . . 7  |-  ( B  =  A  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  if ( ( x  =  A  /\  y  =  A ) ,  A ,  A ) )
11 ifid 3919 . . . . . . 7  |-  if ( ( x  =  A  /\  y  =  A ) ,  A ,  A )  =  A
1210, 11syl6eq 2457 . . . . . 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 2404 . . . . . . . . . . 11  |-  ( y  =  B  ->  (
y  =  A  <->  B  =  A ) )
1514bicomd 201 . . . . . . . . . 10  |-  ( y  =  B  ->  ( B  =  A  <->  y  =  A ) )
1615notbid 292 . . . . . . . . 9  |-  ( y  =  B  ->  ( -.  B  =  A  <->  -.  y  =  A ) )
1716biimpac 484 . . . . . . . 8  |-  ( ( -.  B  =  A  /\  y  =  B )  ->  -.  y  =  A )
1817intnand 915 . . . . . . 7  |-  ( ( -.  B  =  A  /\  y  =  B )  ->  -.  (
x  =  A  /\  y  =  A )
)
1918iffalsed 3893 . . . . . 6  |-  ( ( -.  B  =  A  /\  y  =  B )  ->  if (
( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
2019ex 432 . . . . 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 727 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  ( A  .o.  A )  /\  y  =  B ) )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  A )
23 iftrue 3888 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  B )
2423adantl 464 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  =  A  /\  y  =  A ) )  ->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A )  =  B )
25 simpl 455 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S )  ->  A  e.  S )
26 simpr 459 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  S )  ->  B  e.  S )
279, 24, 25, 25, 26ovmpt2d 6365 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  A
)  =  B )
2827, 26eqeltrd 2488 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A  .o.  A
)  e.  S )
299, 22, 28, 26, 25ovmpt2d 6365 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( ( A  .o.  A )  .o.  B
)  =  A )
303, 5, 29syl2an 475 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 367    = wceq 1403    e. wcel 1840   ifcif 3882   {cpr 3971   ` cfv 5523  (class class class)co 6232    |-> cmpt2 6234   Basecbs 14731   +g cplusg 14799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5487  df-fun 5525  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237
This theorem is referenced by:  mgm2nsgrplem4  16253
  Copyright terms: Public domain W3C validator