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Theorem coahom 16921
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 2760 . . 3 (comp‘𝐶) = (comp‘𝐶)
61, 2, 3, 4, 5coaval 16919 . 2 (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹))⟩)
7 eqid 2760 . . 3 (Base‘𝐶) = (Base‘𝐶)
82homarcl 16879 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
93, 8syl 17 . . 3 (𝜑𝐶 ∈ Cat)
10 eqid 2760 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
112, 7homarcl2 16886 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
123, 11syl 17 . . . 4 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
1312simpld 477 . . 3 (𝜑𝑋 ∈ (Base‘𝐶))
142, 7homarcl2 16886 . . . . 5 (𝐺 ∈ (𝑌𝐻𝑍) → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
154, 14syl 17 . . . 4 (𝜑 → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
1615simprd 482 . . 3 (𝜑𝑍 ∈ (Base‘𝐶))
1712simprd 482 . . . 4 (𝜑𝑌 ∈ (Base‘𝐶))
182, 10homahom 16890 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
193, 18syl 17 . . . 4 (𝜑 → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
202, 10homahom 16890 . . . . 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 16547 . . 3 (𝜑 → ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍))
232, 7, 9, 10, 13, 16, 22elhomai2 16885 . 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 383   = wceq 1632  wcel 2139  cop 4327  cotp 4329  cfv 6049  (class class class)co 6813  2nd c2nd 7332  Basecbs 16059  Hom chom 16154  compcco 16155  Catccat 16526  Homachoma 16874  compaccoa 16905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-ot 4330  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-cat 16530  df-doma 16875  df-coda 16876  df-homa 16877  df-arw 16878  df-coa 16907
This theorem is referenced by:  coapm  16922  arwass  16925
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