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Theorem coahom 17785
Description: The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homdmcoa.o · = (compa𝐶)
homdmcoa.h 𝐻 = (Homa𝐶)
homdmcoa.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
homdmcoa.g (𝜑𝐺 ∈ (𝑌𝐻𝑍))
Assertion
Ref Expression
coahom (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))

Proof of Theorem coahom
StepHypRef Expression
1 homdmcoa.o . . 3 · = (compa𝐶)
2 homdmcoa.h . . 3 𝐻 = (Homa𝐶)
3 homdmcoa.f . . 3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
4 homdmcoa.g . . 3 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
5 eqid 2738 . . 3 (comp‘𝐶) = (comp‘𝐶)
61, 2, 3, 4, 5coaval 17783 . 2 (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹))⟩)
7 eqid 2738 . . 3 (Base‘𝐶) = (Base‘𝐶)
82homarcl 17743 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
93, 8syl 17 . . 3 (𝜑𝐶 ∈ Cat)
10 eqid 2738 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
112, 7homarcl2 17750 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
123, 11syl 17 . . . 4 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
1312simpld 495 . . 3 (𝜑𝑋 ∈ (Base‘𝐶))
142, 7homarcl2 17750 . . . . 5 (𝐺 ∈ (𝑌𝐻𝑍) → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
154, 14syl 17 . . . 4 (𝜑 → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
1615simprd 496 . . 3 (𝜑𝑍 ∈ (Base‘𝐶))
1712simprd 496 . . . 4 (𝜑𝑌 ∈ (Base‘𝐶))
182, 10homahom 17754 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
193, 18syl 17 . . . 4 (𝜑 → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
202, 10homahom 17754 . . . . 5 (𝐺 ∈ (𝑌𝐻𝑍) → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
214, 20syl 17 . . . 4 (𝜑 → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
227, 10, 5, 9, 13, 17, 16, 19, 21catcocl 17394 . . 3 (𝜑 → ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍))
232, 7, 9, 10, 13, 16, 22elhomai2 17749 . 2 (𝜑 → ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹))⟩ ∈ (𝑋𝐻𝑍))
246, 23eqeltrd 2839 1 (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  cop 4567  cotp 4569  cfv 6433  (class class class)co 7275  2nd c2nd 7830  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Homachoma 17738  compaccoa 17769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-ot 4570  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-cat 17377  df-doma 17739  df-coda 17740  df-homa 17741  df-arw 17742  df-coa 17771
This theorem is referenced by:  coapm  17786  arwass  17789
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