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Theorem xpccofval 14997
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 2443 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2443 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
4 eqid 2443 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
5 eqid 2443 . . . 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 14993 . . . . . 6  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
1210, 11eqtr4i 2466 . . . . 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 14994 . . . . 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 2444 . . . 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 14992 . . 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 14872 . . 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 14361 . . 3  |- comp  = Slot  (comp ` 
ndx )
21 snsstp3 4031 . . 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 5706 . . . . . . 7  |-  ( Base `  T )  e.  _V
2310, 22eqeltri 2513 . . . . . 6  |-  B  e. 
_V
2423, 23xpex 6513 . . . . 5  |-  ( B  X.  B )  e. 
_V
2524, 23mpt2ex 6655 . . . 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 14214 . 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 6161 . . . 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 2447 . . 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 14991 . . . . . . . 8  |-  X.c  Fn  ( _V  X.  _V )
32 fndm 5515 . . . . . . . 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 6252 . . . . . 6  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( C  X.c  D )  =  (/) )
351, 34syl5eq 2487 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  T  =  (/) )
3635fveq2d 5700 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  (comp `  T )  =  (comp `  (/) ) )
3720str0 14217 . . . 4  |-  (/)  =  (comp `  (/) )
3836, 27, 373eqtr4g 2500 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  O  =  (/) )
3935fveq2d 5700 . . . . . . 7  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( Base `  T
)  =  ( Base `  (/) ) )
40 base0 14218 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
4139, 10, 403eqtr4g 2500 . . . . . 6  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  B  =  (/) )
4241xpeq2d 4869 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( B  X.  B
)  =  ( B  X.  (/) ) )
43 xp0 5261 . . . . 5  |-  ( B  X.  (/) )  =  (/)
4442, 43syl6eq 2491 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( B  X.  B
)  =  (/) )
45 eqidd 2444 . . . 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 6153 . . 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 2501 . 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 1369    e. wcel 1756   _Vcvv 2977   (/)c0 3642   {ctp 3886   <.cop 3888    X. cxp 4843   dom cdm 4845    Fn wfn 5418   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   1stc1st 6580   2ndc2nd 6581   1c1 9288   5c5 10379  ;cdc 10760   ndxcnx 14176   Basecbs 14179   Hom chom 14254  compcco 14255    X.c cxpc 14983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-hom 14267  df-cco 14268  df-xpc 14987
This theorem is referenced by:  xpcco  14998
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