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

Theorem xpchom 14986
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
)
xpchom.x  |-  ( ph  ->  X  e.  B )
xpchom.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
xpchom  |-  ( ph  ->  ( X K Y )  =  ( ( ( 1st `  X
) H ( 1st `  Y ) )  X.  ( ( 2nd `  X
) J ( 2nd `  Y ) ) ) )

Proof of Theorem xpchom
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpchom.x . 2  |-  ( ph  ->  X  e.  B )
2 xpchom.y . 2  |-  ( ph  ->  Y  e.  B )
3 simpl 454 . . . . . 6  |-  ( ( u  =  X  /\  v  =  Y )  ->  u  =  X )
43fveq2d 5692 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 1st `  u
)  =  ( 1st `  X ) )
5 simpr 458 . . . . . 6  |-  ( ( u  =  X  /\  v  =  Y )  ->  v  =  Y )
65fveq2d 5692 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 1st `  v
)  =  ( 1st `  Y ) )
74, 6oveq12d 6108 . . . 4  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( ( 1st `  u
) H ( 1st `  v ) )  =  ( ( 1st `  X
) H ( 1st `  Y ) ) )
83fveq2d 5692 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 2nd `  u
)  =  ( 2nd `  X ) )
95fveq2d 5692 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 2nd `  v
)  =  ( 2nd `  Y ) )
108, 9oveq12d 6108 . . . 4  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( ( 2nd `  u
) J ( 2nd `  v ) )  =  ( ( 2nd `  X
) J ( 2nd `  Y ) ) )
117, 10xpeq12d 4861 . . 3  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) )  =  ( ( ( 1st `  X ) H ( 1st `  Y ) )  X.  ( ( 2nd `  X ) J ( 2nd `  Y
) ) ) )
12 xpchomfval.t . . . 4  |-  T  =  ( C  X.c  D )
13 xpchomfval.y . . . 4  |-  B  =  ( Base `  T
)
14 xpchomfval.h . . . 4  |-  H  =  ( Hom  `  C
)
15 xpchomfval.j . . . 4  |-  J  =  ( Hom  `  D
)
16 xpchomfval.k . . . 4  |-  K  =  ( Hom  `  T
)
1712, 13, 14, 15, 16xpchomfval 14985 . . 3  |-  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
18 ovex 6115 . . . 4  |-  ( ( 1st `  X ) H ( 1st `  Y
) )  e.  _V
19 ovex 6115 . . . 4  |-  ( ( 2nd `  X ) J ( 2nd `  Y
) )  e.  _V
2018, 19xpex 6507 . . 3  |-  ( ( ( 1st `  X
) H ( 1st `  Y ) )  X.  ( ( 2nd `  X
) J ( 2nd `  Y ) ) )  e.  _V
2111, 17, 20ovmpt2a 6220 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X K Y )  =  ( ( ( 1st `  X
) H ( 1st `  Y ) )  X.  ( ( 2nd `  X
) J ( 2nd `  Y ) ) ) )
221, 2, 21syl2anc 656 1  |-  ( ph  ->  ( X K Y )  =  ( ( ( 1st `  X
) H ( 1st `  Y ) )  X.  ( ( 2nd `  X
) J ( 2nd `  Y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    X. cxp 4834   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   Basecbs 14170   Hom chom 14245    X.c cxpc 14974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-hom 14258  df-cco 14259  df-xpc 14978
This theorem is referenced by:  xpchom2  14992  xpccatid  14994  1stfcl  15003  2ndfcl  15004  xpcpropd  15014  evlfcl  15028  curf1cl  15034  hofcl  15065  yonedalem3  15086
  Copyright terms: Public domain W3C validator