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Theorem fucass 17967
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 2728 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
2 eqid 2728 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
3 eqid 2728 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
4 fucass.r . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐹𝑁𝐺))
5 fucass.n . . . . . . . . . . 11 𝑁 = (𝐶 Nat 𝐷)
65natrcl 17947 . . . . . . . . . 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 493 . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
9 funcrcl 17856 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
108, 9syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1110simprd 494 . . . . . 6 (𝜑𝐷 ∈ Cat)
1211adantr 479 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
13 eqid 2728 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
14 relfunc 17855 . . . . . . . 8 Rel (𝐶 Func 𝐷)
15 1st2ndbr 8052 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1614, 8, 15sylancr 585 . . . . . . 7 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1713, 1, 16funcf1 17859 . . . . . 6 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
1817ffvelcdmda 7099 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
197simprd 494 . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
20 1st2ndbr 8052 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2114, 19, 20sylancr 585 . . . . . . 7 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2213, 1, 21funcf1 17859 . . . . . 6 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
2322ffvelcdmda 7099 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
24 fucass.t . . . . . . . . . 10 (𝜑𝑇 ∈ (𝐻𝑁𝐾))
255natrcl 17947 . . . . . . . . . 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 493 . . . . . . . 8 (𝜑𝐻 ∈ (𝐶 Func 𝐷))
28 1st2ndbr 8052 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
2914, 27, 28sylancr 585 . . . . . . 7 (𝜑 → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
3013, 1, 29funcf1 17859 . . . . . 6 (𝜑 → (1st𝐻):(Base‘𝐶)⟶(Base‘𝐷))
3130ffvelcdmda 7099 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐻)‘𝑥) ∈ (Base‘𝐷))
325, 4nat1st2nd 17948 . . . . . . 7 (𝜑𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
3332adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
34 simpr 483 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
355, 33, 13, 2, 34natcl 17950 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
36 fucass.s . . . . . . . 8 (𝜑𝑆 ∈ (𝐺𝑁𝐻))
375, 36nat1st2nd 17948 . . . . . . 7 (𝜑𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
3837adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
395, 38, 13, 2, 34natcl 17950 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑆𝑥) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐷)((1st𝐻)‘𝑥)))
4026simprd 494 . . . . . . . 8 (𝜑𝐾 ∈ (𝐶 Func 𝐷))
41 1st2ndbr 8052 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐾 ∈ (𝐶 Func 𝐷)) → (1st𝐾)(𝐶 Func 𝐷)(2nd𝐾))
4214, 40, 41sylancr 585 . . . . . . 7 (𝜑 → (1st𝐾)(𝐶 Func 𝐷)(2nd𝐾))
4313, 1, 42funcf1 17859 . . . . . 6 (𝜑 → (1st𝐾):(Base‘𝐶)⟶(Base‘𝐷))
4443ffvelcdmda 7099 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐾)‘𝑥) ∈ (Base‘𝐷))
455, 24nat1st2nd 17948 . . . . . . 7 (𝜑𝑇 ∈ (⟨(1st𝐻), (2nd𝐻)⟩𝑁⟨(1st𝐾), (2nd𝐾)⟩))
4645adantr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑇 ∈ (⟨(1st𝐻), (2nd𝐻)⟩𝑁⟨(1st𝐾), (2nd𝐾)⟩))
475, 46, 13, 2, 34natcl 17950 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑇𝑥) ∈ (((1st𝐻)‘𝑥)(Hom ‘𝐷)((1st𝐾)‘𝑥)))
481, 2, 3, 12, 18, 23, 31, 35, 39, 44, 47catass 17673 . . . 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 17962 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)‘𝑥) = ((𝑇𝑥)(⟨((1st𝐺)‘𝑥), ((1st𝐻)‘𝑥)⟩(comp‘𝐷)((1st𝐾)‘𝑥))(𝑆𝑥)))
5453oveq1d 7441 . . . 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 17962 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑥) = ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐻)‘𝑥))(𝑅𝑥)))
5756oveq2d 7442 . . . 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 2778 . . 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 5252 . 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 17963 . . 3 (𝜑 → (𝑇(⟨𝐺, 𝐻 𝐾)𝑆) ∈ (𝐺𝑁𝐾))
6149, 5, 13, 3, 50, 4, 60fucco 17961 . 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 17963 . . 3 (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
6349, 5, 13, 3, 50, 62, 24fucco 17961 . 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 2778 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 394   = wceq 1533  wcel 2098  cop 4638   class class class wbr 5152  cmpt 5235  Rel wrel 5687  cfv 6553  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  Basecbs 17187  Hom chom 17251  compcco 17252  Catccat 17651   Func cfunc 17847   Nat cnat 17938   FuncCat cfuc 17939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17188  df-hom 17264  df-cco 17265  df-cat 17655  df-func 17851  df-nat 17940  df-fuc 17941
This theorem is referenced by:  fuccatid  17968
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