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Theorem cofuass 17162
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 6106 . . . 4 (((1st𝐾) ∘ (1st𝐻)) ∘ (1st𝐺)) = ((1st𝐾) ∘ ((1st𝐻) ∘ (1st𝐺)))
2 eqid 2824 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
3 cofuass.h . . . . . 6 (𝜑𝐻 ∈ (𝐷 Func 𝐸))
4 cofuass.k . . . . . 6 (𝜑𝐾 ∈ (𝐸 Func 𝐹))
52, 3, 4cofu1st 17156 . . . . 5 (𝜑 → (1st ‘(𝐾func 𝐻)) = ((1st𝐾) ∘ (1st𝐻)))
65coeq1d 5720 . . . 4 (𝜑 → ((1st ‘(𝐾func 𝐻)) ∘ (1st𝐺)) = (((1st𝐾) ∘ (1st𝐻)) ∘ (1st𝐺)))
7 eqid 2824 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
8 cofuass.g . . . . . 6 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
97, 8, 3cofu1st 17156 . . . . 5 (𝜑 → (1st ‘(𝐻func 𝐺)) = ((1st𝐻) ∘ (1st𝐺)))
109coeq2d 5721 . . . 4 (𝜑 → ((1st𝐾) ∘ (1st ‘(𝐻func 𝐺))) = ((1st𝐾) ∘ ((1st𝐻) ∘ (1st𝐺))))
111, 6, 103eqtr4a 2885 . . 3 (𝜑 → ((1st ‘(𝐾func 𝐻)) ∘ (1st𝐺)) = ((1st𝐾) ∘ (1st ‘(𝐻func 𝐺))))
12 coass 6106 . . . . 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 1130 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐻 ∈ (𝐷 Func 𝐸))
1443ad2ant1 1130 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐾 ∈ (𝐸 Func 𝐹))
15 relfunc 17135 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
16 1st2ndbr 7737 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
1715, 8, 16sylancr 590 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
18173ad2ant1 1130 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
197, 2, 18funcf1 17139 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
20 simp2 1134 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
2119, 20ffvelrnd 6844 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
22 simp3 1135 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
2319, 22ffvelrnd 6844 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
242, 13, 14, 21, 23cofu2nd 17158 . . . . . 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 5720 . . . . 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 1130 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐺 ∈ (𝐶 Func 𝐷))
277, 26, 13, 20cofu1 17157 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻func 𝐺))‘𝑥) = ((1st𝐻)‘((1st𝐺)‘𝑥)))
287, 26, 13, 22cofu1 17157 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻func 𝐺))‘𝑦) = ((1st𝐻)‘((1st𝐺)‘𝑦)))
2927, 28oveq12d 7168 . . . . . 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 17158 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘(𝐻func 𝐺))𝑦) = ((((1st𝐺)‘𝑥)(2nd𝐻)((1st𝐺)‘𝑦)) ∘ (𝑥(2nd𝐺)𝑦)))
3129, 30coeq12d 5723 . . . . 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 2885 . . . 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 7225 . . 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 4798 . 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 17161 . . 3 (𝜑 → (𝐾func 𝐻) ∈ (𝐷 Func 𝐹))
367, 8, 35cofuval 17155 . 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 17161 . . 3 (𝜑 → (𝐻func 𝐺) ∈ (𝐶 Func 𝐸))
387, 37, 4cofuval 17155 . 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 2869 1 (𝜑 → ((𝐾func 𝐻) ∘func 𝐺) = (𝐾func (𝐻func 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  cop 4557   class class class wbr 5053  ccom 5547  Rel wrel 5548  cfv 6344  (class class class)co 7150  cmpo 7152  1st c1st 7683  2nd c2nd 7684  Basecbs 16486   Func cfunc 17127  func ccofu 17129
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-map 8405  df-ixp 8459  df-cat 16942  df-cid 16943  df-func 17131  df-cofu 17133
This theorem is referenced by:  catccatid  17365
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