MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  evlf2val Structured version   Unicode version

Theorem evlf2val 14321
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 14320 . 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 734 . . . 4  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
a  =  A )
1514fveq1d 5733 . . 3  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
( a `  Y
)  =  ( A `
 Y ) )
16 simprr 735 . . . 4  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
g  =  K )
1716fveq2d 5735 . . 3  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
( ( X ( 2nd `  F ) Y ) `  g
)  =  ( ( X ( 2nd `  F
) Y ) `  K ) )
1815, 17oveq12d 6102 . 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 6109 . . 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 6204 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   Basecbs 13474    Hom chom 13545  compcco 13546   Catccat 13894    Func cfunc 14056   Nat cnat 14143   evalF cevlf 14311
This theorem is referenced by:  evlfcllem  14323  evlfcl  14324  uncf2  14339  yonedalem3b  14381
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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-reu 2714  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-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-evlf 14315
  Copyright terms: Public domain W3C validator