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Theorem arwass 16924
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 2806 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2806 . . . . 5 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2806 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 arwlid.f . . . . . 6 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
5 arwlid.h . . . . . . 7 𝐻 = (Homa𝐶)
65homarcl 16878 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
74, 6syl 17 . . . . 5 (𝜑𝐶 ∈ Cat)
85, 1homarcl2 16885 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
94, 8syl 17 . . . . . 6 (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
109simpld 484 . . . . 5 (𝜑𝑋 ∈ (Base‘𝐶))
119simprd 485 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
12 arwass.k . . . . . . 7 (𝜑𝐾 ∈ (𝑍𝐻𝑊))
135, 1homarcl2 16885 . . . . . . 7 (𝐾 ∈ (𝑍𝐻𝑊) → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
1412, 13syl 17 . . . . . 6 (𝜑 → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
1514simpld 484 . . . . 5 (𝜑𝑍 ∈ (Base‘𝐶))
165, 2homahom 16889 . . . . . 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 16889 . . . . . 6 (𝐺 ∈ (𝑌𝐻𝑍) → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
2018, 19syl 17 . . . . 5 (𝜑 → (2nd𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
2114simprd 485 . . . . 5 (𝜑𝑊 ∈ (Base‘𝐶))
225, 2homahom 16889 . . . . . 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 16547 . . . 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 16919 . . . . 5 (𝜑 → (2nd ‘(𝐾 · 𝐺)) = ((2nd𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd𝐺)))
2726oveq1d 6885 . . . 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 16919 . . . . 5 (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd𝐹)))
2928oveq2d 6886 . . . 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 2850 . . 3 (𝜑 → ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹)) = ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹))))
3130oteq3d 4609 . 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 16920 . . 3 (𝜑 → (𝐾 · 𝐺) ∈ (𝑌𝐻𝑊))
3325, 5, 4, 32, 3coaval 16918 . 2 (𝜑 → ((𝐾 · 𝐺) · 𝐹) = ⟨𝑋, 𝑊, ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd𝐹))⟩)
3425, 5, 4, 18coahom 16920 . . 3 (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
3525, 5, 34, 12, 3coaval 16918 . 2 (𝜑 → (𝐾 · (𝐺 · 𝐹)) = ⟨𝑋, 𝑊, ((2nd𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹)))⟩)
3631, 33, 353eqtr4d 2850 1 (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  cop 4376  cotp 4378  cfv 6097  (class class class)co 6870  2nd c2nd 7393  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16525  Homachoma 16873  Idacida 16903  compaccoa 16904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-ot 4379  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-1st 7394  df-2nd 7395  df-cat 16529  df-doma 16874  df-coda 16875  df-homa 16876  df-arw 16877  df-coa 16906
This theorem is referenced by: (None)
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