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

Theorem xpchomfval 15111
Description: Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpchomfval.t  |-  T  =  ( C  X.c  D )
xpchomfval.y  |-  B  =  ( Base `  T
)
xpchomfval.h  |-  H  =  ( Hom  `  C
)
xpchomfval.j  |-  J  =  ( Hom  `  D
)
xpchomfval.k  |-  K  =  ( Hom  `  T
)
Assertion
Ref Expression
xpchomfval  |-  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
Distinct variable groups:    v, u, B    u, C, v    u, D, v    u, H, v   
u, J, v
Allowed substitution hints:    T( v, u)    K( v, u)

Proof of Theorem xpchomfval
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpchomfval.t . . . 4  |-  T  =  ( C  X.c  D )
2 eqid 2454 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
3 eqid 2454 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
4 xpchomfval.h . . . 4  |-  H  =  ( Hom  `  C
)
5 xpchomfval.j . . . 4  |-  J  =  ( Hom  `  D
)
6 eqid 2454 . . . 4  |-  (comp `  C )  =  (comp `  C )
7 eqid 2454 . . . 4  |-  (comp `  D )  =  (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 xpchomfval.y . . . . . 6  |-  B  =  ( Base `  T
)
111, 2, 3xpcbas 15110 . . . . . 6  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
1210, 11eqtr4i 2486 . . . . 5  |-  B  =  ( ( Base `  C
)  X.  ( Base `  D ) )
1312a1i 11 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  B  =  ( (
Base `  C )  X.  ( Base `  D
) ) )
14 eqidd 2455 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) ) )
15 eqidd 2455 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) ) y ) ,  f  e.  ( ( u  e.  B , 
v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  C )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) ) y ) ,  f  e.  ( ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  C )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15xpcval 15109 . . 3  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  T  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
>. ,  <. (comp `  ndx ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) y ) ,  f  e.  ( ( u  e.  B , 
v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  C )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } )
17 catstr 14989 . . 3  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
>. ,  <. (comp `  ndx ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) y ) ,  f  e.  ( ( u  e.  B , 
v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  C )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } Struct  <. 1 , ; 1 5 >.
18 homid 14476 . . 3  |-  Hom  = Slot  ( Hom  `  ndx )
19 snsstp2 4136 . . 3  |-  { <. ( Hom  `  ndx ) ,  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
>. }  C_  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
>. ,  <. (comp `  ndx ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) y ) ,  f  e.  ( ( u  e.  B , 
v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  C )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  D )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. }
20 fvex 5812 . . . . . 6  |-  ( Base `  T )  e.  _V
2110, 20eqeltri 2538 . . . . 5  |-  B  e. 
_V
2221, 21mpt2ex 6763 . . . 4  |-  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) )  e.  _V
2322a1i 11 . . 3  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )  e.  _V )
24 xpchomfval.k . . 3  |-  K  =  ( Hom  `  T
)
2516, 17, 18, 19, 23, 24strfv3 14330 . 2  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) )
26 mpt20 6268 . . . 4  |-  ( u  e.  (/) ,  v  e.  (/)  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) )  =  (/)
2726eqcomi 2467 . . 3  |-  (/)  =  ( u  e.  (/) ,  v  e.  (/)  |->  ( ( ( 1st `  u ) H ( 1st `  v
) )  X.  (
( 2nd `  u
) J ( 2nd `  v ) ) ) )
28 fnxpc 15108 . . . . . . . 8  |-  X.c  Fn  ( _V  X.  _V )
29 fndm 5621 . . . . . . . 8  |-  (  X.c  Fn  ( _V  X.  _V )  ->  dom  X.c  =  ( _V  X.  _V ) )
3028, 29ax-mp 5 . . . . . . 7  |-  dom  X.c  =  ( _V  X.  _V )
3130ndmov 6360 . . . . . 6  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( C  X.c  D )  =  (/) )
321, 31syl5eq 2507 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  T  =  (/) )
3332fveq2d 5806 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( Hom  `  T
)  =  ( Hom  `  (/) ) )
3418str0 14333 . . . 4  |-  (/)  =  ( Hom  `  (/) )
3533, 24, 343eqtr4g 2520 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  K  =  (/) )
3632fveq2d 5806 . . . . 5  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( Base `  T
)  =  ( Base `  (/) ) )
37 base0 14334 . . . . 5  |-  (/)  =  (
Base `  (/) )
3836, 10, 373eqtr4g 2520 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  B  =  (/) )
39 eqidd 2455 . . . 4  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) )  =  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
4038, 38, 39mpt2eq123dv 6260 . . 3  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )  =  ( u  e.  (/) ,  v  e.  (/)  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) ) )
4127, 35, 403eqtr4a 2521 . 2  |-  ( -.  ( C  e.  _V  /\  D  e.  _V )  ->  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u
) H ( 1st `  v ) )  X.  ( ( 2nd `  u
) J ( 2nd `  v ) ) ) ) )
4225, 41pm2.61i 164 1  |-  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3748   {ctp 3992   <.cop 3994    X. cxp 4949   dom cdm 4951    Fn wfn 5524   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   1c1 9397   5c5 10488  ;cdc 10869   ndxcnx 14292   Basecbs 14295   Hom chom 14371  compcco 14372    X.c cxpc 15100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-fz 11558  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-hom 14384  df-cco 14385  df-xpc 15104
This theorem is referenced by:  xpchom  15112  relxpchom  15113  xpccofval  15114  catcxpccl  15139  xpcpropd  15140
  Copyright terms: Public domain W3C validator