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Theorem coahom 18024
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 2724 . . 3 (comp‘𝐶) = (comp‘𝐶)
61, 2, 3, 4, 5coaval 18022 . 2 (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹))⟩)
7 eqid 2724 . . 3 (Base‘𝐶) = (Base‘𝐶)
82homarcl 17982 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
93, 8syl 17 . . 3 (𝜑𝐶 ∈ Cat)
10 eqid 2724 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
112, 7homarcl2 17989 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
123, 11syl 17 . . . 4 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
1312simpld 494 . . 3 (𝜑𝑋 ∈ (Base‘𝐶))
142, 7homarcl2 17989 . . . . 5 (𝐺 ∈ (𝑌𝐻𝑍) → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
154, 14syl 17 . . . 4 (𝜑 → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
1615simprd 495 . . 3 (𝜑𝑍 ∈ (Base‘𝐶))
1712simprd 495 . . . 4 (𝜑𝑌 ∈ (Base‘𝐶))
182, 10homahom 17993 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
193, 18syl 17 . . . 4 (𝜑 → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
202, 10homahom 17993 . . . . 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 17630 . . 3 (𝜑 → ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍))
232, 7, 9, 10, 13, 16, 22elhomai2 17988 . 2 (𝜑 → ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹))⟩ ∈ (𝑋𝐻𝑍))
246, 23eqeltrd 2825 1 (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cop 4627  cotp 4629  cfv 6534  (class class class)co 7402  2nd c2nd 7968  Basecbs 17145  Hom chom 17209  compcco 17210  Catccat 17609  Homachoma 17977  compaccoa 18008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-ot 4630  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-cat 17613  df-doma 17978  df-coda 17979  df-homa 17980  df-arw 17981  df-coa 18010
This theorem is referenced by:  coapm  18025  arwass  18028
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