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Theorem sgrp2nmndlem1 16255
Description: Lemma 1 for sgrp2nmnd 16262: 
M is a magma, even if 
A  =  B ( M is the trivial magma in this case, see mgmb1mgm1 16097). (Contributed by AV, 29-Jan-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
) )
Assertion
Ref Expression
sgrp2nmndlem1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  M  e. Mgm )
Distinct variable groups:    x, S, y    x, A, y    x, B, y    x, M
Allowed substitution hints:    M( y)    V( x, y)    W( x, y)

Proof of Theorem sgrp2nmndlem1
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.b . . . 4  |-  ( Base `  M )  =  S
76eqcomi 2413 . . 3  |-  S  =  ( Base `  M
)
8 sgrp2nmnd.o . . 3  |-  ( +g  `  M )  =  ( x  e.  S , 
y  e.  S  |->  if ( x  =  A ,  A ,  B
) )
9 ne0i 3741 . . . 4  |-  ( A  e.  S  ->  S  =/=  (/) )
109adantr 463 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  ->  S  =/=  (/) )
11 simpll 752 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  e.  S  /\  y  e.  S ) )  ->  A  e.  S )
12 simplr 754 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  e.  S  /\  y  e.  S ) )  ->  B  e.  S )
137, 8, 10, 11, 12opifismgm 16099 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  M  e. Mgm )
143, 5, 13syl2an 475 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  M  e. Mgm )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840    =/= wne 2596   (/)c0 3735   ifcif 3882   {cpr 3971   ` cfv 5523    |-> cmpt2 6234   Basecbs 14731   +g cplusg 14799  Mgmcmgm 16084
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-8 1842  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-pow 4569  ax-pr 4627  ax-un 6528
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-csb 3371  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-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-mgm 16086
This theorem is referenced by:  sgrp2nmndlem4  16260
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