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Theorem cofuass 17017
 Description: Functor composition is associative. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofuass.g (𝜑𝐺 ∈ (𝐶 Func 𝐷))
cofuass.h (𝜑𝐻 ∈ (𝐷 Func 𝐸))
cofuass.k (𝜑𝐾 ∈ (𝐸 Func 𝐹))
Assertion
Ref Expression
cofuass (𝜑 → ((𝐾func 𝐻) ∘func 𝐺) = (𝐾func (𝐻func 𝐺)))

Proof of Theorem cofuass
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coass 5957 . . . 4 (((1st𝐾) ∘ (1st𝐻)) ∘ (1st𝐺)) = ((1st𝐾) ∘ ((1st𝐻) ∘ (1st𝐺)))
2 eqid 2778 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
3 cofuass.h . . . . . 6 (𝜑𝐻 ∈ (𝐷 Func 𝐸))
4 cofuass.k . . . . . 6 (𝜑𝐾 ∈ (𝐸 Func 𝐹))
52, 3, 4cofu1st 17011 . . . . 5 (𝜑 → (1st ‘(𝐾func 𝐻)) = ((1st𝐾) ∘ (1st𝐻)))
65coeq1d 5582 . . . 4 (𝜑 → ((1st ‘(𝐾func 𝐻)) ∘ (1st𝐺)) = (((1st𝐾) ∘ (1st𝐻)) ∘ (1st𝐺)))
7 eqid 2778 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
8 cofuass.g . . . . . 6 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
97, 8, 3cofu1st 17011 . . . . 5 (𝜑 → (1st ‘(𝐻func 𝐺)) = ((1st𝐻) ∘ (1st𝐺)))
109coeq2d 5583 . . . 4 (𝜑 → ((1st𝐾) ∘ (1st ‘(𝐻func 𝐺))) = ((1st𝐾) ∘ ((1st𝐻) ∘ (1st𝐺))))
111, 6, 103eqtr4a 2840 . . 3 (𝜑 → ((1st ‘(𝐾func 𝐻)) ∘ (1st𝐺)) = ((1st𝐾) ∘ (1st ‘(𝐻func 𝐺))))
12 coass 5957 . . . . 5 (((((1st𝐻)‘((1st𝐺)‘𝑥))(2nd𝐾)((1st𝐻)‘((1st𝐺)‘𝑦))) ∘ (((1st𝐺)‘𝑥)(2nd𝐻)((1st𝐺)‘𝑦))) ∘ (𝑥(2nd𝐺)𝑦)) = ((((1st𝐻)‘((1st𝐺)‘𝑥))(2nd𝐾)((1st𝐻)‘((1st𝐺)‘𝑦))) ∘ ((((1st𝐺)‘𝑥)(2nd𝐻)((1st𝐺)‘𝑦)) ∘ (𝑥(2nd𝐺)𝑦)))
1333ad2ant1 1113 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐻 ∈ (𝐷 Func 𝐸))
1443ad2ant1 1113 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐾 ∈ (𝐸 Func 𝐹))
15 relfunc 16990 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
16 1st2ndbr 7553 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1715, 8, 16sylancr 578 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
18173ad2ant1 1113 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
197, 2, 18funcf1 16994 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
20 simp2 1117 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
2119, 20ffvelrnd 6677 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
22 simp3 1118 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
2319, 22ffvelrnd 6677 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
242, 13, 14, 21, 23cofu2nd 17013 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st𝐺)‘𝑥)(2nd ‘(𝐾func 𝐻))((1st𝐺)‘𝑦)) = ((((1st𝐻)‘((1st𝐺)‘𝑥))(2nd𝐾)((1st𝐻)‘((1st𝐺)‘𝑦))) ∘ (((1st𝐺)‘𝑥)(2nd𝐻)((1st𝐺)‘𝑦))))
2524coeq1d 5582 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝐺)‘𝑥)(2nd ‘(𝐾func 𝐻))((1st𝐺)‘𝑦)) ∘ (𝑥(2nd𝐺)𝑦)) = (((((1st𝐻)‘((1st𝐺)‘𝑥))(2nd𝐾)((1st𝐻)‘((1st𝐺)‘𝑦))) ∘ (((1st𝐺)‘𝑥)(2nd𝐻)((1st𝐺)‘𝑦))) ∘ (𝑥(2nd𝐺)𝑦)))
2683ad2ant1 1113 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐺 ∈ (𝐶 Func 𝐷))
277, 26, 13, 20cofu1 17012 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻func 𝐺))‘𝑥) = ((1st𝐻)‘((1st𝐺)‘𝑥)))
287, 26, 13, 22cofu1 17012 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻func 𝐺))‘𝑦) = ((1st𝐻)‘((1st𝐺)‘𝑦)))
2927, 28oveq12d 6994 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st ‘(𝐻func 𝐺))‘𝑥)(2nd𝐾)((1st ‘(𝐻func 𝐺))‘𝑦)) = (((1st𝐻)‘((1st𝐺)‘𝑥))(2nd𝐾)((1st𝐻)‘((1st𝐺)‘𝑦))))
307, 26, 13, 20, 22cofu2nd 17013 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘(𝐻func 𝐺))𝑦) = ((((1st𝐺)‘𝑥)(2nd𝐻)((1st𝐺)‘𝑦)) ∘ (𝑥(2nd𝐺)𝑦)))
3129, 30coeq12d 5585 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st ‘(𝐻func 𝐺))‘𝑥)(2nd𝐾)((1st ‘(𝐻func 𝐺))‘𝑦)) ∘ (𝑥(2nd ‘(𝐻func 𝐺))𝑦)) = ((((1st𝐻)‘((1st𝐺)‘𝑥))(2nd𝐾)((1st𝐻)‘((1st𝐺)‘𝑦))) ∘ ((((1st𝐺)‘𝑥)(2nd𝐻)((1st𝐺)‘𝑦)) ∘ (𝑥(2nd𝐺)𝑦))))
3212, 25, 313eqtr4a 2840 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝐺)‘𝑥)(2nd ‘(𝐾func 𝐻))((1st𝐺)‘𝑦)) ∘ (𝑥(2nd𝐺)𝑦)) = ((((1st ‘(𝐻func 𝐺))‘𝑥)(2nd𝐾)((1st ‘(𝐻func 𝐺))‘𝑦)) ∘ (𝑥(2nd ‘(𝐻func 𝐺))𝑦)))
3332mpoeq3dva 7049 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐺)‘𝑥)(2nd ‘(𝐾func 𝐻))((1st𝐺)‘𝑦)) ∘ (𝑥(2nd𝐺)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘(𝐻func 𝐺))‘𝑥)(2nd𝐾)((1st ‘(𝐻func 𝐺))‘𝑦)) ∘ (𝑥(2nd ‘(𝐻func 𝐺))𝑦))))
3411, 33opeq12d 4685 . 2 (𝜑 → ⟨((1st ‘(𝐾func 𝐻)) ∘ (1st𝐺)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐺)‘𝑥)(2nd ‘(𝐾func 𝐻))((1st𝐺)‘𝑦)) ∘ (𝑥(2nd𝐺)𝑦)))⟩ = ⟨((1st𝐾) ∘ (1st ‘(𝐻func 𝐺))), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘(𝐻func 𝐺))‘𝑥)(2nd𝐾)((1st ‘(𝐻func 𝐺))‘𝑦)) ∘ (𝑥(2nd ‘(𝐻func 𝐺))𝑦)))⟩)
353, 4cofucl 17016 . . 3 (𝜑 → (𝐾func 𝐻) ∈ (𝐷 Func 𝐹))
367, 8, 35cofuval 17010 . 2 (𝜑 → ((𝐾func 𝐻) ∘func 𝐺) = ⟨((1st ‘(𝐾func 𝐻)) ∘ (1st𝐺)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝐺)‘𝑥)(2nd ‘(𝐾func 𝐻))((1st𝐺)‘𝑦)) ∘ (𝑥(2nd𝐺)𝑦)))⟩)
378, 3cofucl 17016 . . 3 (𝜑 → (𝐻func 𝐺) ∈ (𝐶 Func 𝐸))
387, 37, 4cofuval 17010 . 2 (𝜑 → (𝐾func (𝐻func 𝐺)) = ⟨((1st𝐾) ∘ (1st ‘(𝐻func 𝐺))), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘(𝐻func 𝐺))‘𝑥)(2nd𝐾)((1st ‘(𝐻func 𝐺))‘𝑦)) ∘ (𝑥(2nd ‘(𝐻func 𝐺))𝑦)))⟩)
3934, 36, 383eqtr4d 2824 1 (𝜑 → ((𝐾func 𝐻) ∘func 𝐺) = (𝐾func (𝐻func 𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050  ⟨cop 4447   class class class wbr 4929   ∘ ccom 5411  Rel wrel 5412  ‘cfv 6188  (class class class)co 6976   ∈ cmpo 6978  1st c1st 7499  2nd c2nd 7500  Basecbs 16339   Func cfunc 16982   ∘func ccofu 16984 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 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279 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 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-map 8208  df-ixp 8260  df-cat 16797  df-cid 16798  df-func 16986  df-cofu 16988 This theorem is referenced by:  catccatid  17220
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