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Theorem arwass 16547
 Description: Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwlid.h 𝐻 = (Homa𝐶)
arwlid.o · = (compa𝐶)
arwlid.a 1 = (Ida𝐶)
arwlid.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
arwass.g (𝜑𝐺 ∈ (𝑌𝐻𝑍))
arwass.k (𝜑𝐾 ∈ (𝑍𝐻𝑊))
Assertion
Ref Expression
arwass (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))

Proof of Theorem arwass
StepHypRef Expression
1 eqid 2610 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2610 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2610 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 arwlid.f . . . . . 6 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
5 arwlid.h . . . . . . 7 𝐻 = (Homa𝐶)
65homarcl 16501 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
74, 6syl 17 . . . . 5 (𝜑𝐶 ∈ Cat)
85, 1homarcl2 16508 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
94, 8syl 17 . . . . . 6 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simpld 474 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
119simprd 478 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
12 arwass.k . . . . . . 7 (𝜑𝐾 ∈ (𝑍𝐻𝑊))
135, 1homarcl2 16508 . . . . . . 7 (𝐾 ∈ (𝑍𝐻𝑊) → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
1412, 13syl 17 . . . . . 6 (𝜑 → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
1514simpld 474 . . . . 5 (𝜑𝑍 ∈ (Base‘𝐶))
165, 2homahom 16512 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
174, 16syl 17 . . . . 5 (𝜑 → (2nd𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
18 arwass.g . . . . . 6 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
195, 2homahom 16512 . . . . . 6 (𝐺 ∈ (𝑌𝐻𝑍) → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
2018, 19syl 17 . . . . 5 (𝜑 → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
2114simprd 478 . . . . 5 (𝜑𝑊 ∈ (Base‘𝐶))
225, 2homahom 16512 . . . . . 6 (𝐾 ∈ (𝑍𝐻𝑊) → (2nd𝐾) ∈ (𝑍(Hom ‘𝐶)𝑊))
2312, 22syl 17 . . . . 5 (𝜑 → (2nd𝐾) ∈ (𝑍(Hom ‘𝐶)𝑊))
241, 2, 3, 7, 10, 11, 15, 17, 20, 21, 23catass 16170 . . . 4 (𝜑 → (((2nd𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹)) = ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹))))
25 arwlid.o . . . . . 6 · = (compa𝐶)
2625, 5, 18, 12, 3coa2 16542 . . . . 5 (𝜑 → (2nd ‘(𝐾 · 𝐺)) = ((2nd𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd𝐺)))
2726oveq1d 6564 . . . 4 (𝜑 → ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹)) = (((2nd𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹)))
2825, 5, 4, 18, 3coa2 16542 . . . . 5 (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹)))
2928oveq2d 6565 . . . 4 (𝜑 → ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹))) = ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹))))
3024, 27, 293eqtr4d 2654 . . 3 (𝜑 → ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹)) = ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹))))
3130oteq3d 4354 . 2 (𝜑 → ⟨𝑋, 𝑊, ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹))⟩ = ⟨𝑋, 𝑊, ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹)))⟩)
3225, 5, 18, 12coahom 16543 . . 3 (𝜑 → (𝐾 · 𝐺) ∈ (𝑌𝐻𝑊))
3325, 5, 4, 32, 3coaval 16541 . 2 (𝜑 → ((𝐾 · 𝐺) · 𝐹) = ⟨𝑋, 𝑊, ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹))⟩)
3425, 5, 4, 18coahom 16543 . . 3 (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
3525, 5, 34, 12, 3coaval 16541 . 2 (𝜑 → (𝐾 · (𝐺 · 𝐹)) = ⟨𝑋, 𝑊, ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹)))⟩)
3631, 33, 353eqtr4d 2654 1 (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  ⟨cotp 4133  ‘cfv 5804  (class class class)co 6549  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Homachoma 16496  Idacida 16526  compaccoa 16527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cat 16152  df-doma 16497  df-coda 16498  df-homa 16499  df-arw 16500  df-coa 16529 This theorem is referenced by: (None)
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