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Theorem fucass 17072
 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 2795 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
2 eqid 2795 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
3 eqid 2795 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
4 fucass.r . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐹𝑁𝐺))
5 fucass.n . . . . . . . . . . 11 𝑁 = (𝐶 Nat 𝐷)
65natrcl 17054 . . . . . . . . . 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 495 . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
9 funcrcl 16967 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
108, 9syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1110simprd 496 . . . . . 6 (𝜑𝐷 ∈ Cat)
1211adantr 481 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
13 eqid 2795 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
14 relfunc 16966 . . . . . . . 8 Rel (𝐶 Func 𝐷)
15 1st2ndbr 7602 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1614, 8, 15sylancr 587 . . . . . . 7 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1713, 1, 16funcf1 16970 . . . . . 6 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
1817ffvelrnda 6721 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
197simprd 496 . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
20 1st2ndbr 7602 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2114, 19, 20sylancr 587 . . . . . . 7 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2213, 1, 21funcf1 16970 . . . . . 6 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
2322ffvelrnda 6721 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
24 fucass.t . . . . . . . . . 10 (𝜑𝑇 ∈ (𝐻𝑁𝐾))
255natrcl 17054 . . . . . . . . . 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 495 . . . . . . . 8 (𝜑𝐻 ∈ (𝐶 Func 𝐷))
28 1st2ndbr 7602 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
2914, 27, 28sylancr 587 . . . . . . 7 (𝜑 → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
3013, 1, 29funcf1 16970 . . . . . 6 (𝜑 → (1st𝐻):(Base‘𝐶)⟶(Base‘𝐷))
3130ffvelrnda 6721 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐻)‘𝑥) ∈ (Base‘𝐷))
325, 4nat1st2nd 17055 . . . . . . 7 (𝜑𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
3332adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
34 simpr 485 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
355, 33, 13, 2, 34natcl 17057 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
36 fucass.s . . . . . . . 8 (𝜑𝑆 ∈ (𝐺𝑁𝐻))
375, 36nat1st2nd 17055 . . . . . . 7 (𝜑𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
3837adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
395, 38, 13, 2, 34natcl 17057 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑆𝑥) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
4026simprd 496 . . . . . . . 8 (𝜑𝐾 ∈ (𝐶 Func 𝐷))
41 1st2ndbr 7602 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐾 ∈ (𝐶 Func 𝐷)) → (1st𝐾)(𝐶 Func 𝐷)(2nd𝐾))
4214, 40, 41sylancr 587 . . . . . . 7 (𝜑 → (1st𝐾)(𝐶 Func 𝐷)(2nd𝐾))
4313, 1, 42funcf1 16970 . . . . . 6 (𝜑 → (1st𝐾):(Base‘𝐶)⟶(Base‘𝐷))
4443ffvelrnda 6721 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐾)‘𝑥) ∈ (Base‘𝐷))
455, 24nat1st2nd 17055 . . . . . . 7 (𝜑𝑇 ∈ (⟨(1st𝐻), (2nd𝐻)⟩𝑁⟨(1st𝐾), (2nd𝐾)⟩))
4645adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑇 ∈ (⟨(1st𝐻), (2nd𝐻)⟩𝑁⟨(1st𝐾), (2nd𝐾)⟩))
475, 46, 13, 2, 34natcl 17057 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑇𝑥) ∈ (((1st𝐻)‘𝑥)(Hom ‘𝐷)((1st𝐾)‘𝑥)))
481, 2, 3, 12, 18, 23, 31, 35, 39, 44, 47catass 16791 . . . 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 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (𝐺𝑁𝐻))
5224adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑇 ∈ (𝐻𝑁𝐾))
5349, 5, 13, 3, 50, 51, 52, 34fuccoval 17067 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)‘𝑥) = ((𝑇𝑥)(⟨((1st𝐺)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑆𝑥)))
5453oveq1d 7036 . . . 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 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (𝐹𝑁𝐺))
5649, 5, 13, 3, 50, 55, 51, 34fuccoval 17067 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥) = ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥)))
5756oveq2d 7037 . . . 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 2841 . . 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 5060 . 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 17068 . . 3 (𝜑 → (𝑇(⟨𝐺, 𝐻 𝐾)𝑆) ∈ (𝐺𝑁𝐾))
6149, 5, 13, 3, 50, 4, 60fucco 17066 . 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 17068 . . 3 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
6349, 5, 13, 3, 50, 62, 24fucco 17066 . 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 2841 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 396   = wceq 1522   ∈ wcel 2081  ⟨cop 4482   class class class wbr 4966   ↦ cmpt 5045  Rel wrel 5453  ‘cfv 6230  (class class class)co 7021  1st c1st 7548  2nd c2nd 7549  Basecbs 16317  Hom chom 16410  compcco 16411  Catccat 16769   Func cfunc 16958   Nat cnat 17045   FuncCat cfuc 17046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-int 4787  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-1st 7550  df-2nd 7551  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-1o 7958  df-oadd 7962  df-er 8144  df-map 8263  df-ixp 8316  df-en 8363  df-dom 8364  df-sdom 8365  df-fin 8366  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-nn 11492  df-2 11553  df-3 11554  df-4 11555  df-5 11556  df-6 11557  df-7 11558  df-8 11559  df-9 11560  df-n0 11751  df-z 11835  df-dec 11953  df-uz 12099  df-fz 12748  df-struct 16319  df-ndx 16320  df-slot 16321  df-base 16323  df-hom 16423  df-cco 16424  df-cat 16773  df-func 16962  df-nat 17047  df-fuc 17048 This theorem is referenced by:  fuccatid  17073
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