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Theorem coahom 17445
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 2739 . . 3 (comp‘𝐶) = (comp‘𝐶)
61, 2, 3, 4, 5coaval 17443 . 2 (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹))⟩)
7 eqid 2739 . . 3 (Base‘𝐶) = (Base‘𝐶)
82homarcl 17403 . . . 4 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
93, 8syl 17 . . 3 (𝜑𝐶 ∈ Cat)
10 eqid 2739 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
112, 7homarcl2 17410 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
123, 11syl 17 . . . 4 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
1312simpld 498 . . 3 (𝜑𝑋 ∈ (Base‘𝐶))
142, 7homarcl2 17410 . . . . 5 (𝐺 ∈ (𝑌𝐻𝑍) → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
154, 14syl 17 . . . 4 (𝜑 → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
1615simprd 499 . . 3 (𝜑𝑍 ∈ (Base‘𝐶))
1712simprd 499 . . . 4 (𝜑𝑌 ∈ (Base‘𝐶))
182, 10homahom 17414 . . . . 5 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
193, 18syl 17 . . . 4 (𝜑 → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
202, 10homahom 17414 . . . . 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 17062 . . 3 (𝜑 → ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍))
232, 7, 9, 10, 13, 16, 22elhomai2 17409 . 2 (𝜑 → ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹))⟩ ∈ (𝑋𝐻𝑍))
246, 23eqeltrd 2834 1 (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  cop 4523  cotp 4525  cfv 6340  (class class class)co 7173  2nd c2nd 7716  Basecbs 16589  Hom chom 16682  compcco 16683  Catccat 17041  Homachoma 17398  compaccoa 17429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-ot 4526  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-1st 7717  df-2nd 7718  df-cat 17045  df-doma 17399  df-coda 17400  df-homa 17401  df-arw 17402  df-coa 17431
This theorem is referenced by:  coapm  17446  arwass  17449
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