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Theorem arwass 16493
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 2609 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2609 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2609 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 arwlid.f . . . . . 6 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
5 arwlid.h . . . . . . 7 𝐻 = (Homa𝐶)
65homarcl 16447 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
74, 6syl 17 . . . . 5 (𝜑𝐶 ∈ Cat)
85, 1homarcl2 16454 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
94, 8syl 17 . . . . . 6 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simpld 473 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
119simprd 477 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
12 arwass.k . . . . . . 7 (𝜑𝐾 ∈ (𝑍𝐻𝑊))
135, 1homarcl2 16454 . . . . . . 7 (𝐾 ∈ (𝑍𝐻𝑊) → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
1412, 13syl 17 . . . . . 6 (𝜑 → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
1514simpld 473 . . . . 5 (𝜑𝑍 ∈ (Base‘𝐶))
165, 2homahom 16458 . . . . . 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 16458 . . . . . 6 (𝐺 ∈ (𝑌𝐻𝑍) → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
2018, 19syl 17 . . . . 5 (𝜑 → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
2114simprd 477 . . . . 5 (𝜑𝑊 ∈ (Base‘𝐶))
225, 2homahom 16458 . . . . . 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 16116 . . . 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 16488 . . . . 5 (𝜑 → (2nd ‘(𝐾 · 𝐺)) = ((2nd𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd𝐺)))
2726oveq1d 6542 . . . 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 16488 . . . . 5 (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹)))
2928oveq2d 6543 . . . 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 2653 . . 3 (𝜑 → ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹)) = ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹))))
3130oteq3d 4348 . 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 16489 . . 3 (𝜑 → (𝐾 · 𝐺) ∈ (𝑌𝐻𝑊))
3325, 5, 4, 32, 3coaval 16487 . 2 (𝜑 → ((𝐾 · 𝐺) · 𝐹) = ⟨𝑋, 𝑊, ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹))⟩)
3425, 5, 4, 18coahom 16489 . . 3 (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
3525, 5, 34, 12, 3coaval 16487 . 2 (𝜑 → (𝐾 · (𝐺 · 𝐹)) = ⟨𝑋, 𝑊, ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹)))⟩)
3631, 33, 353eqtr4d 2653 1 (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cop 4130  cotp 4132  cfv 5790  (class class class)co 6527  2nd c2nd 7035  Basecbs 15641  Hom chom 15725  compcco 15726  Catccat 16094  Homachoma 16442  Idacida 16472  compaccoa 16473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-ot 4133  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-cat 16098  df-doma 16443  df-coda 16444  df-homa 16445  df-arw 16446  df-coa 16475
This theorem is referenced by: (None)
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