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Theorem iscsgrpALT 32812
Description: The predicate "is a commutative semigroup." (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ismgmALT.b  |-  B  =  ( Base `  M
)
ismgmALT.o  |-  .o.  =  ( +g  `  M )
Assertion
Ref Expression
iscsgrpALT  |-  ( M  e. CSGrpALT 
<->  ( M  e. SGrpALT  /\  .o. comLaw  B ) )

Proof of Theorem iscsgrpALT
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . 4  |-  ( m  =  M  ->  ( +g  `  m )  =  ( +g  `  M
) )
2 fveq2 5872 . . . 4  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
31, 2breq12d 4469 . . 3  |-  ( m  =  M  ->  (
( +g  `  m ) comLaw 
( Base `  m )  <->  ( +g  `  M ) comLaw 
( Base `  M )
) )
4 ismgmALT.o . . . 4  |-  .o.  =  ( +g  `  M )
5 ismgmALT.b . . . 4  |-  B  =  ( Base `  M
)
64, 5breq12i 4465 . . 3  |-  (  .o. comLaw  B 
<->  ( +g  `  M
) comLaw  ( Base `  M
) )
73, 6syl6bbr 263 . 2  |-  ( m  =  M  ->  (
( +g  `  m ) comLaw 
( Base `  m )  <->  .o. comLaw  B ) )
8 df-csgrp2 32808 . 2  |- CSGrpALT  =  {
m  e. SGrpALT  |  ( +g  `  m ) comLaw  ( Base `  m ) }
97, 8elrab2 3259 1  |-  ( M  e. CSGrpALT 
<->  ( M  e. SGrpALT  /\  .o. comLaw  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594   Basecbs 14644   +g cplusg 14712   comLaw ccomlaw 32771  SGrpALTcsgrp2 32803  CSGrpALTccsgrp2 32804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-csgrp2 32808
This theorem is referenced by: (None)
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