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Theorem comfffval 15615
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 5870 . . . . . . 7  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
3 comfffval.b . . . . . . 7  |-  B  =  ( Base `  C
)
42, 3syl6eqr 2505 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  B )
54sqxpeqd 4863 . . . . 5  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
6 fveq2 5870 . . . . . . . 8  |-  ( c  =  C  ->  ( Hom  `  c )  =  ( Hom  `  C
) )
7 comfffval.h . . . . . . . 8  |-  H  =  ( Hom  `  C
)
86, 7syl6eqr 2505 . . . . . . 7  |-  ( c  =  C  ->  ( Hom  `  c )  =  H )
98oveqd 6312 . . . . . 6  |-  ( c  =  C  ->  (
( 2nd `  x
) ( Hom  `  c
) y )  =  ( ( 2nd `  x
) H y ) )
108fveq1d 5872 . . . . . 6  |-  ( c  =  C  ->  (
( Hom  `  c ) `
 x )  =  ( H `  x
) )
11 fveq2 5870 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
12 comfffval.x . . . . . . . . 9  |-  .x.  =  (comp `  C )
1311, 12syl6eqr 2505 . . . . . . . 8  |-  ( c  =  C  ->  (comp `  c )  =  .x.  )
1413oveqd 6312 . . . . . . 7  |-  ( c  =  C  ->  (
x (comp `  c
) y )  =  ( x  .x.  y
) )
1514oveqd 6312 . . . . . 6  |-  ( c  =  C  ->  (
g ( x (comp `  c ) y ) f )  =  ( g ( x  .x.  y ) f ) )
169, 10, 15mpt2eq123dv 6358 . . . . 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 6358 . . . 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 15589 . . . 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 5880 . . . . . . 7  |-  ( Base `  C )  e.  _V
203, 19eqeltri 2527 . . . . . 6  |-  B  e. 
_V
2120, 20xpex 6600 . . . . 5  |-  ( B  X.  B )  e. 
_V
2221, 20mpt2ex 6875 . . . 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 5953 . . 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 5864 . . . 4  |-  ( -.  C  e.  _V  ->  (compf `  C )  =  (/) )
25 fvprc 5864 . . . . . . . . 9  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
263, 25syl5eq 2499 . . . . . . . 8  |-  ( -.  C  e.  _V  ->  B  =  (/) )
2726xpeq2d 4861 . . . . . . 7  |-  ( -.  C  e.  _V  ->  ( B  X.  B )  =  ( B  X.  (/) ) )
28 xp0 5258 . . . . . . 7  |-  ( B  X.  (/) )  =  (/)
2927, 28syl6eq 2503 . . . . . 6  |-  ( -.  C  e.  _V  ->  ( B  X.  B )  =  (/) )
30 mpt2eq12 6356 . . . . . 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 667 . . . . 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 6366 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) H y ) ,  f  e.  ( H `  x ) 
|->  ( g ( x 
.x.  y ) f ) ) )  =  (/)
3331, 32syl6eq 2503 . . . 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 2490 . . 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 168 . 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 2475 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 setvar class
Syntax hints:   -. wn 3    = wceq 1446    e. wcel 1889   _Vcvv 3047   (/)c0 3733    X. cxp 4835   ` cfv 5585  (class class class)co 6295    |-> cmpt2 6297   2ndc2nd 6797   Basecbs 15133   Hom chom 15213  compcco 15214  compfccomf 15585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  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 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-comf 15589
This theorem is referenced by:  comffval  15616  comfffval2  15618  comfffn  15621  comfeq  15623
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