Theorem evlf2val

Description: Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017)

	Ref	Expression
Hypotheses	evlfval.e	|- E = ( C evalF D )
	evlfval.c	|- ( ph -> C e. Cat )
	evlfval.d	|- ( ph -> D e. Cat )
	evlfval.b	|- B = ( Base ` C )
	evlfval.h	|- H = ( Hom ` C )
	evlfval.o	|- .x. = ( comp ` D )
	evlfval.n	|- N = ( C Nat D )
	evlf2.f	|- ( ph -> F e. ( C Func D ) )
	evlf2.g	|- ( ph -> G e. ( C Func D ) )
	evlf2.x	|- ( ph -> X e. B )
	evlf2.y	|- ( ph -> Y e. B )
	evlf2.l	|- L = ( <. F , X >. ( 2nd ` E ) <. G , Y >. )
	evlf2val.a	|- ( ph -> A e. ( F N G ) )
	evlf2val.k	|- ( ph -> K e. ( X H Y ) )
Assertion	evlf2val	|- ( ph -> ( A L K ) = ( ( A ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) )

Proof

Step	Hyp	Ref	Expression
1		evlfval.e	|- E = ( C evalF D )
2		evlfval.c	|- ( ph -> C e. Cat )
3		evlfval.d	|- ( ph -> D e. Cat )
4		evlfval.b	|- B = ( Base ` C )
5		evlfval.h	|- H = ( Hom ` C )
6		evlfval.o	|- .x. = ( comp ` D )
7		evlfval.n	|- N = ( C Nat D )
8		evlf2.f	|- ( ph -> F e. ( C Func D ) )
9		evlf2.g	|- ( ph -> G e. ( C Func D ) )
10		evlf2.x	|- ( ph -> X e. B )
11		evlf2.y	|- ( ph -> Y e. B )
12		evlf2.l	|- L = ( <. F , X >. ( 2nd ` E ) <. G , Y >. )
13		evlf2val.a	|- ( ph -> A e. ( F N G ) )
14		evlf2val.k	|- ( ph -> K e. ( X H Y ) )
15	1 2 3 4 5 6 7 8 9 10 11 12	evlf2	|- ( ph -> L = ( a e. ( F N G ) , g e. ( X H Y ) |-> ( ( a ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` g ) ) ) )
16		simprl	|- ( ( ph /\ ( a = A /\ g = K ) ) -> a = A )
17	16	fveq1d	|- ( ( ph /\ ( a = A /\ g = K ) ) -> ( a ` Y ) = ( A ` Y ) )
18		simprr	|- ( ( ph /\ ( a = A /\ g = K ) ) -> g = K )
19	18	fveq2d	|- ( ( ph /\ ( a = A /\ g = K ) ) -> ( ( X ( 2nd ` F ) Y ) ` g ) = ( ( X ( 2nd ` F ) Y ) ` K ) )
20	17 19	oveq12d	|- ( ( ph /\ ( a = A /\ g = K ) ) -> ( ( a ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` g ) ) = ( ( A ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) )
21		ovexd	|- ( ph -> ( ( A ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) e. _V )
22	15 20 13 14 21	ovmpod	|- ( ph -> ( A L K ) = ( ( A ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` K ) ) )