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

Theorem comfffval 13929
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffval.o  |-  O  =  (compf `  C )
comfffval.b  |-  B  =  ( Base `  C
)
comfffval.h  |-  H  =  (  Hom  `  C
)
comfffval.x  |-  .x.  =  (comp `  C )
Assertion
Ref Expression
comfffval  |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) )
Distinct variable groups:    x, y, B    f, g, x, y, C    .x. , f, g, x   
f, H, g, x
Allowed substitution hints:    B( f, g)    .x. ( y)    H( y)    O( x, y, f, g)

Proof of Theorem comfffval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 comfffval.o . 2  |-  O  =  (compf `  C )
2 fveq2 5731 . . . . . . 7  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
3 comfffval.b . . . . . . 7  |-  B  =  ( Base `  C
)
42, 3syl6eqr 2488 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  B )
54, 4xpeq12d 4906 . . . . 5  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
6 fveq2 5731 . . . . . . . 8  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
7 comfffval.h . . . . . . . 8  |-  H  =  (  Hom  `  C
)
86, 7syl6eqr 2488 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  =  H )
98oveqd 6101 . . . . . 6  |-  ( c  =  C  ->  (
( 2nd `  x
) (  Hom  `  c
) y )  =  ( ( 2nd `  x
) H y ) )
108fveq1d 5733 . . . . . 6  |-  ( c  =  C  ->  (
(  Hom  `  c ) `
 x )  =  ( H `  x
) )
11 fveq2 5731 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
12 comfffval.x . . . . . . . . 9  |-  .x.  =  (comp `  C )
1311, 12syl6eqr 2488 . . . . . . . 8  |-  ( c  =  C  ->  (comp `  c )  =  .x.  )
1413oveqd 6101 . . . . . . 7  |-  ( c  =  C  ->  (
x (comp `  c
) y )  =  ( x  .x.  y
) )
1514oveqd 6101 . . . . . 6  |-  ( c  =  C  ->  (
g ( x (comp `  c ) y ) f )  =  ( g ( x  .x.  y ) f ) )
169, 10, 15mpt2eq123dv 6139 . . . . 5  |-  ( c  =  C  ->  (
g  e.  ( ( 2nd `  x ) (  Hom  `  c
) y ) ,  f  e.  ( (  Hom  `  c ) `  x )  |->  ( g ( x (comp `  c ) y ) f ) )  =  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) )
175, 4, 16mpt2eq123dv 6139 . . . 4  |-  ( c  =  C  ->  (
x  e.  ( (
Base `  c )  X.  ( Base `  c
) ) ,  y  e.  ( Base `  c
)  |->  ( g  e.  ( ( 2nd `  x
) (  Hom  `  c
) y ) ,  f  e.  ( (  Hom  `  c ) `  x )  |->  ( g ( x (comp `  c ) y ) f ) ) )  =  ( x  e.  ( B  X.  B
) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) ) )
18 df-comf 13901 . . . 4  |- compf  =  ( c  e. 
_V  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) ) ,  y  e.  ( Base `  c )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  c ) y ) ,  f  e.  ( (  Hom  `  c
) `  x )  |->  ( g ( x (comp `  c )
y ) f ) ) ) )
19 fvex 5745 . . . . . . 7  |-  ( Base `  C )  e.  _V
203, 19eqeltri 2508 . . . . . 6  |-  B  e. 
_V
2120, 20xpex 4993 . . . . 5  |-  ( B  X.  B )  e. 
_V
2221, 20mpt2ex 6428 . . . 4  |-  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `  x
)  |->  ( g ( x  .x.  y ) f ) ) )  e.  _V
2317, 18, 22fvmpt 5809 . . 3  |-  ( C  e.  _V  ->  (compf `  C
)  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `  x
)  |->  ( g ( x  .x.  y ) f ) ) ) )
24 fvprc 5725 . . . 4  |-  ( -.  C  e.  _V  ->  (compf `  C )  =  (/) )
25 fvprc 5725 . . . . . . . . 9  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
263, 25syl5eq 2482 . . . . . . . 8  |-  ( -.  C  e.  _V  ->  B  =  (/) )
2726xpeq2d 4905 . . . . . . 7  |-  ( -.  C  e.  _V  ->  ( B  X.  B )  =  ( B  X.  (/) ) )
28 xp0 5294 . . . . . . 7  |-  ( B  X.  (/) )  =  (/)
2927, 28syl6eq 2486 . . . . . 6  |-  ( -.  C  e.  _V  ->  ( B  X.  B )  =  (/) )
30 mpt2eq12 6137 . . . . . 6  |-  ( ( ( B  X.  B
)  =  (/)  /\  B  =  (/) )  ->  (
x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `
 x )  |->  ( g ( x  .x.  y ) f ) ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) ) )
3129, 26, 30syl2anc 644 . . . . 5  |-  ( -.  C  e.  _V  ->  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `
 x )  |->  ( g ( x  .x.  y ) f ) ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) ) )
32 mpt20 6430 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) )  =  (/)
3331, 32syl6eq 2486 . . . 4  |-  ( -.  C  e.  _V  ->  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `
 x )  |->  ( g ( x  .x.  y ) f ) ) )  =  (/) )
3424, 33eqtr4d 2473 . . 3  |-  ( -.  C  e.  _V  ->  (compf `  C )  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `
 x )  |->  ( g ( x  .x.  y ) f ) ) ) )
3523, 34pm2.61i 159 . 2  |-  (compf `  C
)  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) H y ) ,  f  e.  ( H `  x
)  |->  ( g ( x  .x.  y ) f ) ) )
361, 35eqtri 2458 1  |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630    X. cxp 4879   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   2ndc2nd 6351   Basecbs 13474    Hom chom 13545  compcco 13546  compfccomf 13897
This theorem is referenced by:  comffval  13930  comfffval2  13932  comfffn  13935  comfeq  13937
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-comf 13901
  Copyright terms: Public domain W3C validator