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Theorem arwass 17405
 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 2758 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2758 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2758 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 arwlid.f . . . . . 6 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
5 arwlid.h . . . . . . 7 𝐻 = (Homa𝐶)
65homarcl 17359 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
74, 6syl 17 . . . . 5 (𝜑𝐶 ∈ Cat)
85, 1homarcl2 17366 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
94, 8syl 17 . . . . . 6 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simpld 498 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
119simprd 499 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
12 arwass.k . . . . . . 7 (𝜑𝐾 ∈ (𝑍𝐻𝑊))
135, 1homarcl2 17366 . . . . . . 7 (𝐾 ∈ (𝑍𝐻𝑊) → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
1412, 13syl 17 . . . . . 6 (𝜑 → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
1514simpld 498 . . . . 5 (𝜑𝑍 ∈ (Base‘𝐶))
165, 2homahom 17370 . . . . . 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 17370 . . . . . 6 (𝐺 ∈ (𝑌𝐻𝑍) → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
2018, 19syl 17 . . . . 5 (𝜑 → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
2114simprd 499 . . . . 5 (𝜑𝑊 ∈ (Base‘𝐶))
225, 2homahom 17370 . . . . . 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 17020 . . . 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 17400 . . . . 5 (𝜑 → (2nd ‘(𝐾 · 𝐺)) = ((2nd𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd𝐺)))
2726oveq1d 7170 . . . 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 17400 . . . . 5 (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹)))
2928oveq2d 7171 . . . 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 2803 . . 3 (𝜑 → ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹)) = ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹))))
3130oteq3d 4780 . 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 17401 . . 3 (𝜑 → (𝐾 · 𝐺) ∈ (𝑌𝐻𝑊))
3325, 5, 4, 32, 3coaval 17399 . 2 (𝜑 → ((𝐾 · 𝐺) · 𝐹) = ⟨𝑋, 𝑊, ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹))⟩)
3425, 5, 4, 18coahom 17401 . . 3 (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
3525, 5, 34, 12, 3coaval 17399 . 2 (𝜑 → (𝐾 · (𝐺 · 𝐹)) = ⟨𝑋, 𝑊, ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹)))⟩)
3631, 33, 353eqtr4d 2803 1 (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4531  ⟨cotp 4533  ‘cfv 6339  (class class class)co 7155  2nd c2nd 7697  Basecbs 16546  Hom chom 16639  compcco 16640  Catccat 16998  Homachoma 17354  Idacida 17384  compaccoa 17385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-ot 4534  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7698  df-2nd 7699  df-cat 17002  df-doma 17355  df-coda 17356  df-homa 17357  df-arw 17358  df-coa 17387 This theorem is referenced by: (None)
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