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Theorem evlf2val 14271
Description: Value of the evaluation natural transformation at an object. (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 )
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
Ref Expression
evlf2val  |-  ( ph  ->  ( A L K )  =  ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) ) )

Proof of Theorem evlf2val
Dummy variables  a 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlfval.e . . 3  |-  E  =  ( C evalF  D )
2 evlfval.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 evlfval.d . . 3  |-  ( ph  ->  D  e.  Cat )
4 evlfval.b . . 3  |-  B  =  ( Base `  C
)
5 evlfval.h . . 3  |-  H  =  (  Hom  `  C
)
6 evlfval.o . . 3  |-  .x.  =  (comp `  D )
7 evlfval.n . . 3  |-  N  =  ( C Nat  D )
8 evlf2.f . . 3  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
9 evlf2.g . . 3  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
10 evlf2.x . . 3  |-  ( ph  ->  X  e.  B )
11 evlf2.y . . 3  |-  ( ph  ->  Y  e.  B )
12 evlf2.l . . 3  |-  L  =  ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12evlf2 14270 . 2  |-  ( 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 ) ) ) )
14 simprl 733 . . . 4  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
a  =  A )
1514fveq1d 5689 . . 3  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
( a `  Y
)  =  ( A `
 Y ) )
16 simprr 734 . . . 4  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
g  =  K )
1716fveq2d 5691 . . 3  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
( ( X ( 2nd `  F ) Y ) `  g
)  =  ( ( X ( 2nd `  F
) Y ) `  K ) )
1815, 17oveq12d 6058 . 2  |-  ( (
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 ) ) )
19 evlf2val.a . 2  |-  ( ph  ->  A  e.  ( F N G ) )
20 evlf2val.k . 2  |-  ( ph  ->  K  e.  ( X H Y ) )
21 ovex 6065 . . 3  |-  ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) )  e. 
_V
2221a1i 11 . 2  |-  ( ph  ->  ( ( A `  Y ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >.  .x.  ( ( 1st `  G ) `  Y ) ) ( ( X ( 2nd `  F ) Y ) `
 K ) )  e.  _V )
2313, 18, 19, 20, 22ovmpt2d 6160 1  |-  ( ph  ->  ( A L K )  =  ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844    Func cfunc 14006   Nat cnat 14093   evalF cevlf 14261
This theorem is referenced by:  evlfcllem  14273  evlfcl  14274  uncf2  14289  yonedalem3b  14331
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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-reu 2673  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-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-evlf 14265
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