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

Theorem xpchom 14232
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 444 . . . . . 6  |-  ( ( u  =  X  /\  v  =  Y )  ->  u  =  X )
43fveq2d 5691 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 1st `  u
)  =  ( 1st `  X ) )
5 simpr 448 . . . . . 6  |-  ( ( u  =  X  /\  v  =  Y )  ->  v  =  Y )
65fveq2d 5691 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 1st `  v
)  =  ( 1st `  Y ) )
74, 6oveq12d 6058 . . . 4  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( ( 1st `  u
) H ( 1st `  v ) )  =  ( ( 1st `  X
) H ( 1st `  Y ) ) )
83fveq2d 5691 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 2nd `  u
)  =  ( 2nd `  X ) )
95fveq2d 5691 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 2nd `  v
)  =  ( 2nd `  Y ) )
108, 9oveq12d 6058 . . . 4  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( ( 2nd `  u
) J ( 2nd `  v ) )  =  ( ( 2nd `  X
) J ( 2nd `  Y ) ) )
117, 10xpeq12d 4862 . . 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 14231 . . 3  |-  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
18 ovex 6065 . . . 4  |-  ( ( 1st `  X ) H ( 1st `  Y
) )  e.  _V
19 ovex 6065 . . . 4  |-  ( ( 2nd `  X ) J ( 2nd `  Y
) )  e.  _V
2018, 19xpex 4949 . . 3  |-  ( ( ( 1st `  X
) H ( 1st `  Y ) )  X.  ( ( 2nd `  X
) J ( 2nd `  Y ) ) )  e.  _V
2111, 17, 20ovmpt2a 6163 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X K Y )  =  ( ( ( 1st `  X
) H ( 1st `  Y ) )  X.  ( ( 2nd `  X
) J ( 2nd `  Y ) ) ) )
221, 2, 21syl2anc 643 1  |-  ( ph  ->  ( X K Y )  =  ( ( ( 1st `  X
) H ( 1st `  Y ) )  X.  ( ( 2nd `  X
) J ( 2nd `  Y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    X. cxp 4835   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   Basecbs 13424    Hom chom 13495    X.c cxpc 14220
This theorem is referenced by:  xpchom2  14238  xpccatid  14240  1stfcl  14249  2ndfcl  14250  xpcpropd  14260  evlfcl  14274  curf1cl  14280  hofcl  14311  yonedalem3  14332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-hom 13508  df-cco 13509  df-xpc 14224
  Copyright terms: Public domain W3C validator