Theorem iscsgrpALT

Description: The predicate "is a commutative semigroup". (Contributed by AV, 20-Jan-2020) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	ismgmALT.b	$$⊢B={\mathrm{Base}}_M$$
	ismgmALT.o	No typesetting found for |- .o. = ( +g ` M ) with typecode |-
Assertion	iscsgrpALT	Could not format assertion : No typesetting found for |- ( M e. CSGrpALT <-> ( M e. SGrpALT /\ .o. comLaw B ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		ismgmALT.b	$$⊢B={\mathrm{Base}}_M$$
2		ismgmALT.o	Could not format .o. = ( +g ` M ) : No typesetting found for |- .o. = ( +g ` M ) with typecode |-
3		fveq2	$$⊢m=M\to {+}_m={+}_M$$
4		fveq2	$$⊢m=M\to {\mathrm{Base}}_m={\mathrm{Base}}_M$$
5	3 4	breq12d	$$⊢m=M\to \left({+}_m\mathrm{comLaw}{\mathrm{Base}}_m\leftrightarrow {+}_M\mathrm{comLaw}{\mathrm{Base}}_M\right)$$
6	2 1	breq12i	Could not format ( .o. comLaw B <-> ( +g ` M ) comLaw ( Base ` M ) ) : No typesetting found for |- ( .o. comLaw B <-> ( +g ` M ) comLaw ( Base ` M ) ) with typecode |-
7	5 6	bitr4di	Could not format ( m = M -> ( ( +g ` m ) comLaw ( Base ` m ) <-> .o. comLaw B ) ) : No typesetting found for |- ( m = M -> ( ( +g ` m ) comLaw ( Base ` m ) <-> .o. comLaw B ) ) with typecode |-
8		df-csgrp2	$$⊢\mathrm{CSGrpALT}=\left\{m\in \mathrm{SGrpALT}|{+}_m\mathrm{comLaw}{\mathrm{Base}}_m\right\}$$
9	7 8	elrab2	Could not format ( M e. CSGrpALT <-> ( M e. SGrpALT /\ .o. comLaw B ) ) : No typesetting found for |- ( M e. CSGrpALT <-> ( M e. SGrpALT /\ .o. comLaw B ) ) with typecode |-