Theorem iscsgrpALT 41862

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 6191	. . . 4 |- ( m = M -> ( +g ` m ) = ( +g ` M ) )
2		fveq2 6191	. . . 4 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
3	1, 2	breq12d 4666	. . 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 4662	. . 3 |- ( .o. comLaw B <-> ( +g ` M ) comLaw ( Base ` M ) )
7	3, 6	syl6bbr 278	. 2 |- ( m = M -> ( ( +g ` m ) comLaw ( Base ` m ) <-> .o. comLaw B ) )
8		df-csgrp2 41858	. 2 |- CSGrpALT = { m e. SGrpALT | ( +g ` m ) comLaw ( Base ` m ) }
9	7, 8	elrab2 3366	1 |- ( M e. CSGrpALT <-> ( M e. SGrpALT /\ .o. comLaw B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888   Basecbs 15857   +g cplusg 15941   comLaw ccomlaw 41821  SGrpALTcsgrp2 41853  CSGrpALTccsgrp2 41854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-csgrp2 41858

This theorem is referenced by: (None)