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Theorem fucass 16831
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 2771 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
2 eqid 2771 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
3 eqid 2771 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
4 fucass.r . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐹𝑁𝐺))
5 fucass.n . . . . . . . . . . 11 𝑁 = (𝐶 Nat 𝐷)
65natrcl 16813 . . . . . . . . . 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 482 . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
9 funcrcl 16726 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
108, 9syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1110simprd 483 . . . . . 6 (𝜑𝐷 ∈ Cat)
1211adantr 466 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
13 eqid 2771 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
14 relfunc 16725 . . . . . . . 8 Rel (𝐶 Func 𝐷)
15 1st2ndbr 7366 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1614, 8, 15sylancr 575 . . . . . . 7 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1713, 1, 16funcf1 16729 . . . . . 6 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
1817ffvelrnda 6501 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
197simprd 483 . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
20 1st2ndbr 7366 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2114, 19, 20sylancr 575 . . . . . . 7 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2213, 1, 21funcf1 16729 . . . . . 6 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
2322ffvelrnda 6501 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
24 fucass.t . . . . . . . . . 10 (𝜑𝑇 ∈ (𝐻𝑁𝐾))
255natrcl 16813 . . . . . . . . . 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 482 . . . . . . . 8 (𝜑𝐻 ∈ (𝐶 Func 𝐷))
28 1st2ndbr 7366 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
2914, 27, 28sylancr 575 . . . . . . 7 (𝜑 → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
3013, 1, 29funcf1 16729 . . . . . 6 (𝜑 → (1st𝐻):(Base‘𝐶)⟶(Base‘𝐷))
3130ffvelrnda 6501 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐻)‘𝑥) ∈ (Base‘𝐷))
325, 4nat1st2nd 16814 . . . . . . 7 (𝜑𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
3332adantr 466 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
34 simpr 471 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
355, 33, 13, 2, 34natcl 16816 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
36 fucass.s . . . . . . . 8 (𝜑𝑆 ∈ (𝐺𝑁𝐻))
375, 36nat1st2nd 16814 . . . . . . 7 (𝜑𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
3837adantr 466 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
395, 38, 13, 2, 34natcl 16816 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑆𝑥) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
4026simprd 483 . . . . . . . 8 (𝜑𝐾 ∈ (𝐶 Func 𝐷))
41 1st2ndbr 7366 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐾 ∈ (𝐶 Func 𝐷)) → (1st𝐾)(𝐶 Func 𝐷)(2nd𝐾))
4214, 40, 41sylancr 575 . . . . . . 7 (𝜑 → (1st𝐾)(𝐶 Func 𝐷)(2nd𝐾))
4313, 1, 42funcf1 16729 . . . . . 6 (𝜑 → (1st𝐾):(Base‘𝐶)⟶(Base‘𝐷))
4443ffvelrnda 6501 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐾)‘𝑥) ∈ (Base‘𝐷))
455, 24nat1st2nd 16814 . . . . . . 7 (𝜑𝑇 ∈ (⟨(1st𝐻), (2nd𝐻)⟩𝑁⟨(1st𝐾), (2nd𝐾)⟩))
4645adantr 466 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑇 ∈ (⟨(1st𝐻), (2nd𝐻)⟩𝑁⟨(1st𝐾), (2nd𝐾)⟩))
475, 46, 13, 2, 34natcl 16816 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑇𝑥) ∈ (((1st𝐻)‘𝑥)(Hom ‘𝐷)((1st𝐾)‘𝑥)))
481, 2, 3, 12, 18, 23, 31, 35, 39, 44, 47catass 16550 . . . 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 466 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (𝐺𝑁𝐻))
5224adantr 466 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑇 ∈ (𝐻𝑁𝐾))
5349, 5, 13, 3, 50, 51, 52, 34fuccoval 16826 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)‘𝑥) = ((𝑇𝑥)(⟨((1st𝐺)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑆𝑥)))
5453oveq1d 6807 . . . 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 466 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (𝐹𝑁𝐺))
5649, 5, 13, 3, 50, 55, 51, 34fuccoval 16826 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥) = ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥)))
5756oveq2d 6808 . . . 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 2815 . . 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 4878 . 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 16827 . . 3 (𝜑 → (𝑇(⟨𝐺, 𝐻 𝐾)𝑆) ∈ (𝐺𝑁𝐾))
6149, 5, 13, 3, 50, 4, 60fucco 16825 . 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 16827 . . 3 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
6349, 5, 13, 3, 50, 62, 24fucco 16825 . 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 2815 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 1631  wcel 2145  cop 4322   class class class wbr 4786  cmpt 4863  Rel wrel 5254  cfv 6029  (class class class)co 6792  1st c1st 7313  2nd c2nd 7314  Basecbs 16060  Hom chom 16156  compcco 16157  Catccat 16528   Func cfunc 16717   Nat cnat 16804   FuncCat cfuc 16805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11496  df-z 11581  df-dec 11697  df-uz 11890  df-fz 12530  df-struct 16062  df-ndx 16063  df-slot 16064  df-base 16066  df-hom 16170  df-cco 16171  df-cat 16532  df-func 16721  df-nat 16806  df-fuc 16807
This theorem is referenced by:  fuccatid  16832
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