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Theorem arwass 17182
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 2772 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2772 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2772 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 arwlid.f . . . . . 6 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
5 arwlid.h . . . . . . 7 𝐻 = (Homa𝐶)
65homarcl 17136 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
74, 6syl 17 . . . . 5 (𝜑𝐶 ∈ Cat)
85, 1homarcl2 17143 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
94, 8syl 17 . . . . . 6 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simpld 487 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
119simprd 488 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
12 arwass.k . . . . . . 7 (𝜑𝐾 ∈ (𝑍𝐻𝑊))
135, 1homarcl2 17143 . . . . . . 7 (𝐾 ∈ (𝑍𝐻𝑊) → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
1412, 13syl 17 . . . . . 6 (𝜑 → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
1514simpld 487 . . . . 5 (𝜑𝑍 ∈ (Base‘𝐶))
165, 2homahom 17147 . . . . . 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 17147 . . . . . 6 (𝐺 ∈ (𝑌𝐻𝑍) → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
2018, 19syl 17 . . . . 5 (𝜑 → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
2114simprd 488 . . . . 5 (𝜑𝑊 ∈ (Base‘𝐶))
225, 2homahom 17147 . . . . . 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 16805 . . . 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 17177 . . . . 5 (𝜑 → (2nd ‘(𝐾 · 𝐺)) = ((2nd𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd𝐺)))
2726oveq1d 6985 . . . 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 17177 . . . . 5 (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹)))
2928oveq2d 6986 . . . 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 2818 . . 3 (𝜑 → ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹)) = ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹))))
3130oteq3d 4685 . 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 17178 . . 3 (𝜑 → (𝐾 · 𝐺) ∈ (𝑌𝐻𝑊))
3325, 5, 4, 32, 3coaval 17176 . 2 (𝜑 → ((𝐾 · 𝐺) · 𝐹) = ⟨𝑋, 𝑊, ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹))⟩)
3425, 5, 4, 18coahom 17178 . . 3 (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
3525, 5, 34, 12, 3coaval 17176 . 2 (𝜑 → (𝐾 · (𝐺 · 𝐹)) = ⟨𝑋, 𝑊, ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹)))⟩)
3631, 33, 353eqtr4d 2818 1 (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2048  cop 4441  cotp 4443  cfv 6182  (class class class)co 6970  2nd c2nd 7493  Basecbs 16329  Hom chom 16422  compcco 16423  Catccat 16783  Homachoma 17131  Idacida 17161  compaccoa 17162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-ot 4444  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-cat 16787  df-doma 17132  df-coda 17133  df-homa 17134  df-arw 17135  df-coa 17164
This theorem is referenced by: (None)
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