Theorem evlfval 14319

Description: Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
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 )

Assertion

Ref

Expression

evlfval

|- ( ph -> E = <. ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. )

Distinct variable groups:   f, a, g, m, n, x, y, C   D, a, f, g, m, n, x, y   g, H, m, n, x, y   N, a, g, m, n, x, y   ph, a, f, g, m, n, x, y   .x. , a, g, m, n, x, y   x, B, y

Allowed substitution hints:   B( f, g, m, n, a)   .x. ( f)   E( x, y, f, g, m, n, a)   H( f, a)   N( f)

Proof of Theorem evlfval

Dummy variables c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlfval.e	. 2 |- E = ( C evalF D )
2		df-evlf 14315	. . . 4 |- evalF = ( c e. Cat , d e. Cat |-> <. ( f e. ( c Func d ) , x e. ( Base ` c ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( c Func d ) X. ( Base ` c ) ) , y e. ( ( c Func d ) X. ( Base ` c ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( c Nat d ) n ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` d ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. )
3	2	a1i 11	. . 3 |- ( ph -> evalF = ( c e. Cat , d e. Cat |-> <. ( f e. ( c Func d ) , x e. ( Base ` c ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( c Func d ) X. ( Base ` c ) ) , y e. ( ( c Func d ) X. ( Base ` c ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( c Nat d ) n ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` d ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. ) )
4		simprl 734	. . . . . 6 |- ( ( ph /\ ( c = C /\ d = D ) ) -> c = C )
5		simprr 735	. . . . . 6 |- ( ( ph /\ ( c = C /\ d = D ) ) -> d = D )
6	4, 5	oveq12d 6102	. . . . 5 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( c Func d ) = ( C Func D ) )
7	4	fveq2d 5735	. . . . . 6 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( Base ` c ) = ( Base ` C ) )
8		evlfval.b	. . . . . 6 |- B = ( Base ` C )
9	7, 8	syl6eqr 2488	. . . . 5 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( Base ` c ) = B )
10		eqidd 2439	. . . . 5 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( ( 1st ` f ) ` x ) = ( ( 1st ` f ) ` x ) )
11	6, 9, 10	mpt2eq123dv 6139	. . . 4 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( f e. ( c Func d ) , x e. ( Base ` c ) |-> ( ( 1st ` f ) ` x ) ) = ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) )
12	6, 9	xpeq12d 4906	. . . . 5 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( ( c Func d ) X. ( Base ` c ) ) = ( ( C Func D ) X. B ) )
13	4, 5	oveq12d 6102	. . . . . . . . . 10 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( c Nat d ) = ( C Nat D ) )
14		evlfval.n	. . . . . . . . . 10 |- N = ( C Nat D )
15	13, 14	syl6eqr 2488	. . . . . . . . 9 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( c Nat d ) = N )
16	15	oveqd 6101	. . . . . . . 8 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( m ( c Nat d ) n ) = ( m N n ) )
17	4	fveq2d 5735	. . . . . . . . . 10 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( Hom ` c ) = ( Hom ` C ) )
18		evlfval.h	. . . . . . . . . 10 |- H = ( Hom ` C )
19	17, 18	syl6eqr 2488	. . . . . . . . 9 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( Hom ` c ) = H )
20	19	oveqd 6101	. . . . . . . 8 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) = ( ( 2nd ` x ) H ( 2nd ` y ) ) )
21	5	fveq2d 5735	. . . . . . . . . . 11 |- ( ( ph /\ ( c = C /\ d = D ) ) -> (comp ` d ) = (comp ` D ) )
22		evlfval.o	. . . . . . . . . . 11 |- .x. = (comp ` D )
23	21, 22	syl6eqr 2488	. . . . . . . . . 10 |- ( ( ph /\ ( c = C /\ d = D ) ) -> (comp ` d ) = .x. )
24	23	oveqd 6101	. . . . . . . . 9 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` d ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) = ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) )
25	24	oveqd 6101	. . . . . . . 8 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` d ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) = ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) )
26	16, 20, 25	mpt2eq123dv 6139	. . . . . . 7 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( a e. ( m ( c Nat d ) n ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` d ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) = ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) )
27	26	csbeq2dv 3278	. . . . . 6 |- ( ( ph /\ ( c = C /\ d = D ) ) -> [_ ( 1st ` y ) / n ]_ ( a e. ( m ( c Nat d ) n ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` d ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) = [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) )
28	27	csbeq2dv 3278	. . . . 5 |- ( ( ph /\ ( c = C /\ d = D ) ) -> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( c Nat d ) n ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` d ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) = [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) )
29	12, 12, 28	mpt2eq123dv 6139	. . . 4 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( x e. ( ( c Func d ) X. ( Base ` c ) ) , y e. ( ( c Func d ) X. ( Base ` c ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( c Nat d ) n ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` d ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) = ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) )
30	11, 29	opeq12d 3994	. . 3 |- ( ( ph /\ ( c = C /\ d = D ) ) -> <. ( f e. ( c Func d ) , x e. ( Base ` c ) |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( c Func d ) X. ( Base ` c ) ) , y e. ( ( c Func d ) X. ( Base ` c ) ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( c Nat d ) n ) , g e. ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` d ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. = <. ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. )
31		evlfval.c	. . 3 |- ( ph -> C e. Cat )
32		evlfval.d	. . 3 |- ( ph -> D e. Cat )
33		opex 4430	. . . 4 |- <. ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. e. _V
34	33	a1i 11	. . 3 |- ( ph -> <. ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. e. _V )
35	3, 30, 31, 32, 34	ovmpt2d 6204	. 2 |- ( ph -> ( C evalF D ) = <. ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. )
36	1, 35	syl5eq 2482	1 |- ( ph -> E = <. ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. )

Syntax hints:   -> wi 4   /\ wa 360   = wceq 1653   e. wcel 1726   _Vcvv 2958   [_csb 3253   <.cop 3819   X. cxp 4879   ` cfv 5457  (class class class)co 6084   e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351   Basecbs 13474   Hom chom 13545  compcco 13546   Catccat 13894   Func cfunc 14056   Nat cnat 14143   eval_F cevlf 14311

This theorem is referenced by:  evlf2  14320  evlf1  14322  evlfcl  14324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-evlf 14315