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.

Step	Hyp	Ref	Expression
1		fveq2 5872	. . . 4 |- ( m = M -> ( +g ` m ) = ( +g ` M ) )
2		fveq2 5872	. . . 4 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
3	1, 2	breq12d 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 )
6	4, 5	breq12i 4465	. . 3 |- ( .o. comLaw B <-> ( +g ` M ) comLaw ( Base ` M ) )
7	3, 6	syl6bbr 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 ) }
9	7, 8	elrab2 3259	1 |- ( M e. CSGrpALT <-> ( M e. SGrpALT /\ .o. comLaw B ) )

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)