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Theorem xpccofval 15302
Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpccofval.t  |-  T  =  ( C  X.c  D )
xpccofval.b  |-  B  =  ( Base `  T
)
xpccofval.k  |-  K  =  ( Hom  `  T
)
xpccofval.o1  |-  .x.  =  (comp `  C )
xpccofval.o2  |-  .xb  =  (comp `  D )
xpccofval.o  |-  O  =  (comp `  T )
Assertion
Ref Expression
xpccofval  |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )
Distinct variable groups:    f, g, x, y, B    C, f,
g, x, y    D, f, g, x, y    .x. , f,
g, x, y    .xb , f,
g, x, y    f, K, g, x, y    x, O, y
Allowed substitution hints:    T( x, y, f, g)    O( f, g)

Proof of Theorem xpccofval
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpccofval.t . . . 4  |-  T  =  ( C  X.c  D )
2 eqid 2467 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2467 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
4 eqid 2467 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
5 eqid 2467 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
6 xpccofval.o1 . . . 4  |-  .x.  =  (comp `  C )
7 xpccofval.o2 . . . 4  |-  .xb  =  (comp `  D )
8 simpl 457 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  C  e.  _V )
9 simpr 461 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  D  e.  _V )
10 xpccofval.b . . . . . 6  |-  B  =  ( Base `  T
)
111, 2, 3xpcbas 15298 . . . . . 6  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
1210, 11eqtr4i 2499 . . . . 5  |-  B  =  ( ( Base `  C
)  X.  ( Base `  D ) )
1312a1i 11 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  B  =  ( (
Base `  C )  X.  ( Base `  D
) ) )
14 xpccofval.k . . . . . 6  |-  K  =  ( Hom  `  T
)
151, 10, 4, 5, 14xpchomfval 15299 . . . . 5  |-  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) ( Hom  `  C ) ( 1st `  v ) )  X.  ( ( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) )
1615a1i 11 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) ( Hom  `  C
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  D
) ( 2nd `  v
) ) ) ) )
17 eqidd 2468 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
181, 2, 3, 4, 5, 6, 7, 8, 9, 13, 16, 17xpcval 15297 . . 3  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  T  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  K >. ,  <. (comp ` 
ndx ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } )
19 catstr 15177 . . 3  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  K >. ,  <. (comp ` 
ndx ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } Struct  <. 1 , ; 1 5 >.
20 ccoid 14666 . . 3  |- comp  = Slot  (comp ` 
ndx )
21 snsstp3 4180 . . 3  |-  { <. (comp `  ndx ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. }  C_  {
<. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  K >. , 
<. (comp `  ndx ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. }
22 fvex 5874 . . . . . . 7  |-  ( Base `  T )  e.  _V
2310, 22eqeltri 2551 . . . . . 6  |-  B  e. 
_V
2423, 23xpex 6711 . . . . 5  |-  ( B  X.  B )  e. 
_V
2524, 23mpt2ex 6857 . . . 4  |-  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `  x
)  |->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  e.  _V
2625a1i 11 . . 3  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  e.  _V )
27 xpccofval.o . . 3  |-  O  =  (comp `  T )
2818, 19, 20, 21, 26, 27strfv3 14518 . 2  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `  x
)  |->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
29 mpt20 6349 . . . 4  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  (/)
3029eqcomi 2480 . . 3  |-  (/)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )
31 fnxpc 15296 . . . . . . . 8  |-  X.c  Fn  ( _V  X.  _V )
32 fndm 5678 . . . . . . . 8  |-  (  X.c  Fn  ( _V  X.  _V )  ->  dom  X.c  =  ( _V  X.  _V ) )
3331, 32ax-mp 5 . . . . . . 7  |-  dom  X.c  =  ( _V  X.  _V )
3433ndmov 6441 . . . . . 6  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( C  X.c  D )  =  (/) )
351, 34syl5eq 2520 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  T  =  (/) )
3635fveq2d 5868 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  (comp `  T )  =  (comp `  (/) ) )
3720str0 14521 . . . 4  |-  (/)  =  (comp `  (/) )
3836, 27, 373eqtr4g 2533 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  O  =  (/) )
3935fveq2d 5868 . . . . . . 7  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( Base `  T
)  =  ( Base `  (/) ) )
40 base0 14522 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
4139, 10, 403eqtr4g 2533 . . . . . 6  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  B  =  (/) )
4241xpeq2d 5023 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( B  X.  B
)  =  ( B  X.  (/) ) )
43 xp0 5423 . . . . 5  |-  ( B  X.  (/) )  =  (/)
4442, 43syl6eq 2524 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( B  X.  B
)  =  (/) )
45 eqidd 2468 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  =  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `  x
)  |->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )
4644, 41, 45mpt2eq123dv 6341 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
4730, 38, 463eqtr4a 2534 . 2  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `  x
)  |->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
4828, 47pm2.61i 164 1  |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   {ctp 4031   <.cop 4033    X. cxp 4997   dom cdm 4999    Fn wfn 5581   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   1c1 9489   5c5 10584  ;cdc 10972   ndxcnx 14480   Basecbs 14483   Hom chom 14559  compcco 14560    X.c cxpc 15288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-hom 14572  df-cco 14573  df-xpc 15292
This theorem is referenced by:  xpcco  15303
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