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Theorem mgm2nsgrplem1 16015
Description: Lemma 1 for mgm2nsgrp 16019: 
M is a magma, even if 
A  =  B ( M is the trivial magma in this case, see mgmb1mgm1 15862). (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 ) )
Assertion
Ref Expression
mgm2nsgrplem1  |-  ( ( 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 mgm2nsgrplem1
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.b . . . 4  |-  ( Base `  M )  =  S
76eqcomi 2456 . . 3  |-  S  =  ( Base `  M
)
8 mgm2nsgrp.o . . 3  |-  ( +g  `  M )  =  ( x  e.  S , 
y  e.  S  |->  if ( ( x  =  A  /\  y  =  A ) ,  B ,  A ) )
9 ne0i 3776 . . . 4  |-  ( A  e.  S  ->  S  =/=  (/) )
109adantr 465 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  ->  S  =/=  (/) )
11 simplr 755 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  e.  S  /\  y  e.  S ) )  ->  B  e.  S )
12 simpll 753 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( x  e.  S  /\  y  e.  S ) )  ->  A  e.  S )
137, 8, 10, 11, 12opifismgm 15864 . 2  |-  ( ( A  e.  S  /\  B  e.  S )  ->  M  e. Mgm )
143, 5, 13syl2an 477 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  M  e. Mgm )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   (/)c0 3770   ifcif 3926   {cpr 4016   ` cfv 5578    |-> cmpt2 6283   Basecbs 14614   +g cplusg 14679  Mgmcmgm 15849
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-8 1806  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-pow 4615  ax-pr 4676  ax-un 6577
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-csb 3421  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-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-mgm 15851
This theorem is referenced by:  mgm2nsgrp  16019  mgmnsgrpex  16028
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