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Theorem iscsgrpALT 39915
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 5865 . . . 4  |-  ( m  =  M  ->  ( +g  `  m )  =  ( +g  `  M
) )
2 fveq2 5865 . . . 4  |-  ( m  =  M  ->  ( Base `  m )  =  ( Base `  M
) )
31, 2breq12d 4415 . . 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 4411 . . 3  |-  (  .o. comLaw  B 
<->  ( +g  `  M
) comLaw  ( Base `  M
) )
73, 6syl6bbr 267 . 2  |-  ( m  =  M  ->  (
( +g  `  m ) comLaw 
( Base `  m )  <->  .o. comLaw  B ) )
8 df-csgrp2 39911 . 2  |- CSGrpALT  =  {
m  e. SGrpALT  |  ( +g  `  m ) comLaw  ( Base `  m ) }
97, 8elrab2 3198 1  |-  ( M  e. CSGrpALT 
<->  ( M  e. SGrpALT  /\  .o. comLaw  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   class class class wbr 4402   ` cfv 5582   Basecbs 15121   +g cplusg 15190   comLaw ccomlaw 39874  SGrpALTcsgrp2 39906  CSGrpALTccsgrp2 39907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-iota 5546  df-fv 5590  df-csgrp2 39911
This theorem is referenced by: (None)
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