Theorem mndtcco2

Description: The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024)

	Ref	Expression
Hypotheses	mndtcbas.c	|- ( ph -> C = ( MndToCat ` M ) )
	mndtcbas.m	|- ( ph -> M e. Mnd )
	mndtcbas.b	|- ( ph -> B = ( Base ` C ) )
	mndtchom.x	|- ( ph -> X e. B )
	mndtchom.y	|- ( ph -> Y e. B )
	mndtcco.z	|- ( ph -> Z e. B )
	mndtcco.o	|- ( ph -> .x. = ( comp ` C ) )
	mndtcco2.o2	|- ( ph -> .o. = ( <. X , Y >. .x. Z ) )
Assertion	mndtcco2	|- ( ph -> ( G .o. F ) = ( G ( +g ` M ) F ) )

Proof

Step	Hyp	Ref	Expression
1		mndtcbas.c	|- ( ph -> C = ( MndToCat ` M ) )
2		mndtcbas.m	|- ( ph -> M e. Mnd )
3		mndtcbas.b	|- ( ph -> B = ( Base ` C ) )
4		mndtchom.x	|- ( ph -> X e. B )
5		mndtchom.y	|- ( ph -> Y e. B )
6		mndtcco.z	|- ( ph -> Z e. B )
7		mndtcco.o	|- ( ph -> .x. = ( comp ` C ) )
8		mndtcco2.o2	|- ( ph -> .o. = ( <. X , Y >. .x. Z ) )
9	1 2 3 4 5 6 7	mndtcco	|- ( ph -> ( <. X , Y >. .x. Z ) = ( +g ` M ) )
10	8 9	eqtrd	|- ( ph -> .o. = ( +g ` M ) )
11	10	oveqd	|- ( ph -> ( G .o. F ) = ( G ( +g ` M ) F ) )