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

Theorem xpchom 18236
Description: Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpchomfval.t 𝑇 = (𝐶 ×c 𝐷)
xpchomfval.y 𝐵 = (Base‘𝑇)
xpchomfval.h 𝐻 = (Hom ‘𝐶)
xpchomfval.j 𝐽 = (Hom ‘𝐷)
xpchomfval.k 𝐾 = (Hom ‘𝑇)
xpchom.x (𝜑𝑋𝐵)
xpchom.y (𝜑𝑌𝐵)
Assertion
Ref Expression
xpchom (𝜑 → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))

Proof of Theorem xpchom
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpchom.x . 2 (𝜑𝑋𝐵)
2 xpchom.y . 2 (𝜑𝑌𝐵)
3 simpl 482 . . . . . 6 ((𝑢 = 𝑋𝑣 = 𝑌) → 𝑢 = 𝑋)
43fveq2d 6911 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (1st𝑢) = (1st𝑋))
5 simpr 484 . . . . . 6 ((𝑢 = 𝑋𝑣 = 𝑌) → 𝑣 = 𝑌)
65fveq2d 6911 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (1st𝑣) = (1st𝑌))
74, 6oveq12d 7449 . . . 4 ((𝑢 = 𝑋𝑣 = 𝑌) → ((1st𝑢)𝐻(1st𝑣)) = ((1st𝑋)𝐻(1st𝑌)))
83fveq2d 6911 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (2nd𝑢) = (2nd𝑋))
95fveq2d 6911 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (2nd𝑣) = (2nd𝑌))
108, 9oveq12d 7449 . . . 4 ((𝑢 = 𝑋𝑣 = 𝑌) → ((2nd𝑢)𝐽(2nd𝑣)) = ((2nd𝑋)𝐽(2nd𝑌)))
117, 10xpeq12d 5720 . . 3 ((𝑢 = 𝑋𝑣 = 𝑌) → (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
12 xpchomfval.t . . . 4 𝑇 = (𝐶 ×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 18235 . . 3 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
18 ovex 7464 . . . 4 ((1st𝑋)𝐻(1st𝑌)) ∈ V
19 ovex 7464 . . . 4 ((2nd𝑋)𝐽(2nd𝑌)) ∈ V
2018, 19xpex 7772 . . 3 (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))) ∈ V
2111, 17, 20ovmpoa 7588 . 2 ((𝑋𝐵𝑌𝐵) → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
221, 2, 21syl2anc 584 1 (𝜑 → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106   × cxp 5687  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  Hom chom 17309   ×c cxpc 18224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-xpc 18228
This theorem is referenced by:  xpchom2  18242  xpccatid  18244  1stfcl  18253  2ndfcl  18254  xpcpropd  18265  evlfcl  18279  curf1cl  18285  hofcl  18316  yonedalem3  18337
  Copyright terms: Public domain W3C validator