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.

Step	Hyp	Ref	Expression
1		fveq2 5865	. . . 4 |- ( m = M -> ( +g ` m ) = ( +g ` M ) )
2		fveq2 5865	. . . 4 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
3	1, 2	breq12d 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 )
6	4, 5	breq12i 4411	. . 3 |- ( .o. comLaw B <-> ( +g ` M ) comLaw ( Base ` M ) )
7	3, 6	syl6bbr 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 ) }
9	7, 8	elrab2 3198	1 |- ( M e. CSGrpALT <-> ( M e. SGrpALT /\ .o. comLaw B ) )

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)