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Theorem fucass 16397
Description: Associativity of natural transformation composition. Remark 6.14(b) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucass.q 𝑄 = (𝐶 FuncCat 𝐷)
fucass.n 𝑁 = (𝐶 Nat 𝐷)
fucass.x = (comp‘𝑄)
fucass.r (𝜑𝑅 ∈ (𝐹𝑁𝐺))
fucass.s (𝜑𝑆 ∈ (𝐺𝑁𝐻))
fucass.t (𝜑𝑇 ∈ (𝐻𝑁𝐾))
Assertion
Ref Expression
fucass (𝜑 → ((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)(⟨𝐹, 𝐺 𝐾)𝑅) = (𝑇(⟨𝐹, 𝐻 𝐾)(𝑆(⟨𝐹, 𝐺 𝐻)𝑅)))

Proof of Theorem fucass
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
2 eqid 2609 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
3 eqid 2609 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
4 fucass.r . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐹𝑁𝐺))
5 fucass.n . . . . . . . . . . 11 𝑁 = (𝐶 Nat 𝐷)
65natrcl 16379 . . . . . . . . . 10 (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
74, 6syl 17 . . . . . . . . 9 (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
87simpld 473 . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
9 funcrcl 16292 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
108, 9syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1110simprd 477 . . . . . 6 (𝜑𝐷 ∈ Cat)
1211adantr 479 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
13 eqid 2609 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
14 relfunc 16291 . . . . . . . 8 Rel (𝐶 Func 𝐷)
15 1st2ndbr 7085 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1614, 8, 15sylancr 693 . . . . . . 7 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1713, 1, 16funcf1 16295 . . . . . 6 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
1817ffvelrnda 6252 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
197simprd 477 . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
20 1st2ndbr 7085 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2114, 19, 20sylancr 693 . . . . . . 7 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2213, 1, 21funcf1 16295 . . . . . 6 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
2322ffvelrnda 6252 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
24 fucass.t . . . . . . . . . 10 (𝜑𝑇 ∈ (𝐻𝑁𝐾))
255natrcl 16379 . . . . . . . . . 10 (𝑇 ∈ (𝐻𝑁𝐾) → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝐾 ∈ (𝐶 Func 𝐷)))
2624, 25syl 17 . . . . . . . . 9 (𝜑 → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝐾 ∈ (𝐶 Func 𝐷)))
2726simpld 473 . . . . . . . 8 (𝜑𝐻 ∈ (𝐶 Func 𝐷))
28 1st2ndbr 7085 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
2914, 27, 28sylancr 693 . . . . . . 7 (𝜑 → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
3013, 1, 29funcf1 16295 . . . . . 6 (𝜑 → (1st𝐻):(Base‘𝐶)⟶(Base‘𝐷))
3130ffvelrnda 6252 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐻)‘𝑥) ∈ (Base‘𝐷))
325, 4nat1st2nd 16380 . . . . . . 7 (𝜑𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
3332adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
34 simpr 475 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
355, 33, 13, 2, 34natcl 16382 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
36 fucass.s . . . . . . . 8 (𝜑𝑆 ∈ (𝐺𝑁𝐻))
375, 36nat1st2nd 16380 . . . . . . 7 (𝜑𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
3837adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
395, 38, 13, 2, 34natcl 16382 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑆𝑥) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
4026simprd 477 . . . . . . . 8 (𝜑𝐾 ∈ (𝐶 Func 𝐷))
41 1st2ndbr 7085 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐾 ∈ (𝐶 Func 𝐷)) → (1st𝐾)(𝐶 Func 𝐷)(2nd𝐾))
4214, 40, 41sylancr 693 . . . . . . 7 (𝜑 → (1st𝐾)(𝐶 Func 𝐷)(2nd𝐾))
4313, 1, 42funcf1 16295 . . . . . 6 (𝜑 → (1st𝐾):(Base‘𝐶)⟶(Base‘𝐷))
4443ffvelrnda 6252 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐾)‘𝑥) ∈ (Base‘𝐷))
455, 24nat1st2nd 16380 . . . . . . 7 (𝜑𝑇 ∈ (⟨(1st𝐻), (2nd𝐻)⟩𝑁⟨(1st𝐾), (2nd𝐾)⟩))
4645adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑇 ∈ (⟨(1st𝐻), (2nd𝐻)⟩𝑁⟨(1st𝐾), (2nd𝐾)⟩))
475, 46, 13, 2, 34natcl 16382 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑇𝑥) ∈ (((1st𝐻)‘𝑥)(Hom ‘𝐷)((1st𝐾)‘𝑥)))
481, 2, 3, 12, 18, 23, 31, 35, 39, 44, 47catass 16116 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (((𝑇𝑥)(⟨((1st𝐺)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑆𝑥))(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑅𝑥)) = ((𝑇𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥))))
49 fucass.q . . . . . 6 𝑄 = (𝐶 FuncCat 𝐷)
50 fucass.x . . . . . 6 = (comp‘𝑄)
5136adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (𝐺𝑁𝐻))
5224adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑇 ∈ (𝐻𝑁𝐾))
5349, 5, 13, 3, 50, 51, 52, 34fuccoval 16392 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)‘𝑥) = ((𝑇𝑥)(⟨((1st𝐺)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑆𝑥)))
5453oveq1d 6542 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑅𝑥)) = (((𝑇𝑥)(⟨((1st𝐺)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑆𝑥))(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑅𝑥)))
554adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (𝐹𝑁𝐺))
5649, 5, 13, 3, 50, 55, 51, 34fuccoval 16392 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥) = ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥)))
5756oveq2d 6543 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑇𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥)) = ((𝑇𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥))))
5848, 54, 573eqtr4d 2653 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → (((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑅𝑥)) = ((𝑇𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥)))
5958mpteq2dva 4666 . 2 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ (((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑅𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑇𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥))))
6049, 5, 50, 36, 24fuccocl 16393 . . 3 (𝜑 → (𝑇(⟨𝐺, 𝐻 𝐾)𝑆) ∈ (𝐺𝑁𝐾))
6149, 5, 13, 3, 50, 4, 60fucco 16391 . 2 (𝜑 → ((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)(⟨𝐹, 𝐺 𝐾)𝑅) = (𝑥 ∈ (Base‘𝐶) ↦ (((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑅𝑥))))
6249, 5, 50, 4, 36fuccocl 16393 . . 3 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
6349, 5, 13, 3, 50, 62, 24fucco 16391 . 2 (𝜑 → (𝑇(⟨𝐹, 𝐻 𝐾)(𝑆(⟨𝐹, 𝐺 𝐻)𝑅)) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑇𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥))))
6459, 61, 633eqtr4d 2653 1 (𝜑 → ((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)(⟨𝐹, 𝐺 𝐾)𝑅) = (𝑇(⟨𝐹, 𝐻 𝐾)(𝑆(⟨𝐹, 𝐺 𝐻)𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cop 4130   class class class wbr 4577  cmpt 4637  Rel wrel 5033  cfv 5790  (class class class)co 6527  1st c1st 7034  2nd c2nd 7035  Basecbs 15641  Hom chom 15725  compcco 15726  Catccat 16094   Func cfunc 16283   Nat cnat 16370   FuncCat cfuc 16371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-fz 12153  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-hom 15739  df-cco 15740  df-cat 16098  df-func 16287  df-nat 16372  df-fuc 16373
This theorem is referenced by:  fuccatid  16398
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