Theorem evlfval 14269

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 14265	. . . 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 733	. . . . . 6 |- ( ( ph /\ ( c = C /\ d = D ) ) -> c = C )
5		simprr 734	. . . . . 6 |- ( ( ph /\ ( c = C /\ d = D ) ) -> d = D )
6	4, 5	oveq12d 6058	. . . . 5 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( c Func d ) = ( C Func D ) )
7	4	fveq2d 5691	. . . . . 6 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( Base ` c ) = ( Base ` C ) )
8		evlfval.b	. . . . . 6 |- B = ( Base ` C )
9	7, 8	syl6eqr 2454	. . . . 5 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( Base ` c ) = B )
10		eqidd 2405	. . . . 5 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( ( 1st ` f ) ` x ) = ( ( 1st ` f ) ` x ) )
11	6, 9, 10	mpt2eq123dv 6095	. . . 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 4862	. . . . 5 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( ( c Func d ) X. ( Base ` c ) ) = ( ( C Func D ) X. B ) )
13	4, 5	oveq12d 6058	. . . . . . . . . 10 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( c Nat d ) = ( C Nat D ) )
14		evlfval.n	. . . . . . . . . 10 |- N = ( C Nat D )
15	13, 14	syl6eqr 2454	. . . . . . . . 9 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( c Nat d ) = N )
16	15	oveqd 6057	. . . . . . . 8 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( m ( c Nat d ) n ) = ( m N n ) )
17	4	fveq2d 5691	. . . . . . . . . 10 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( Hom ` c ) = ( Hom ` C ) )
18		evlfval.h	. . . . . . . . . 10 |- H = ( Hom ` C )
19	17, 18	syl6eqr 2454	. . . . . . . . 9 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( Hom ` c ) = H )
20	19	oveqd 6057	. . . . . . . 8 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( ( 2nd ` x ) ( Hom ` c ) ( 2nd ` y ) ) = ( ( 2nd ` x ) H ( 2nd ` y ) ) )
21	5	fveq2d 5691	. . . . . . . . . . 11 |- ( ( ph /\ ( c = C /\ d = D ) ) -> (comp ` d ) = (comp ` D ) )
22		evlfval.o	. . . . . . . . . . 11 |- .x. = (comp ` D )
23	21, 22	syl6eqr 2454	. . . . . . . . . 10 |- ( ( ph /\ ( c = C /\ d = D ) ) -> (comp ` d ) = .x. )
24	23	oveqd 6057	. . . . . . . . 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 6057	. . . . . . . 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 6095	. . . . . . 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 3236	. . . . . 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 3236	. . . . 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 6095	. . . 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 3952	. . 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 4387	. . . 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 6160	. 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 2448	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 359   = wceq 1649   e. wcel 1721   _Vcvv 2916   [_csb 3211   <.cop 3777   X. cxp 4835   ` cfv 5413  (class class class)co 6040   e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   Basecbs 13424   Hom chom 13495  compcco 13496   Catccat 13844   Func cfunc 14006   Nat cnat 14093   eval_F cevlf 14261

This theorem is referenced by:  evlf2  14270  evlf1  14272  evlfcl  14274

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-evlf 14265