Theorem natoppf 49719

Description: A natural transformation is natural between opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)

Hypotheses

Ref	Expression
natoppf.o	⊢ 𝑂 = (oppCat‘𝐶)
natoppf.p	⊢ 𝑃 = (oppCat‘𝐷)
natoppf.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
natoppf.m	⊢ 𝑀 = (𝑂 Nat 𝑃)
natoppf.a	⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))

Assertion

Ref	Expression
natoppf	⊢ (𝜑 → 𝐴 ∈ (⟨𝐾, tpos 𝐿⟩𝑀⟨𝐹, tpos 𝐺⟩))

Proof of Theorem natoppf

Dummy variables 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		natoppf.m	. 2 ⊢ 𝑀 = (𝑂 Nat 𝑃)
2		natoppf.o	. . 3 ⊢ 𝑂 = (oppCat‘𝐶)
3		eqid 2739	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4	2, 3	oppcbas 17675	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
5		eqid 2739	. 2 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
6		eqid 2739	. 2 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
7		eqid 2739	. 2 ⊢ (comp‘𝑃) = (comp‘𝑃)
8		natoppf.p	. . 3 ⊢ 𝑃 = (oppCat‘𝐷)
9		natoppf.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
10		natoppf.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))
11	9, 10	natrcl3 49715	. . 3 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)
12	2, 8, 11	funcoppc 17833	. 2 ⊢ (𝜑 → 𝐾(𝑂 Func 𝑃)tpos 𝐿)
13	9, 10	natrcl2 49714	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
14	2, 8, 13	funcoppc 17833	. 2 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)
15	9, 10, 3	natfn 17915	. 2 ⊢ (𝜑 → 𝐴 Fn (Base‘𝐶))
16	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))
17		eqid 2739	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
19	9, 16, 3, 17, 18	natcl 17914	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥)))
20	17, 8	oppchom 17672	. . 3 ⊢ ((𝐾‘𝑥)(Hom ‘𝑃)(𝐹‘𝑥)) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥))
21	19, 20	eleqtrrdi 2850	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ ((𝐾‘𝑥)(Hom ‘𝑃)(𝐹‘𝑥)))
22	10	ad2antrr 732	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))
23		eqid 2739	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
24		eqid 2739	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
25		simplrr 783	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑦 ∈ (Base‘𝐶))
26		simplrl 782	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑥 ∈ (Base‘𝐶))
27		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦))
28	23, 2	oppchom 17672	. . . . . 6 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
29	27, 28	eleqtrdi 2849	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑚 ∈ (𝑦(Hom ‘𝐶)𝑥))
30	9, 22, 3, 23, 24, 25, 26, 29	nati 17916	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → ((𝐴‘𝑥)(⟨(𝐹‘𝑦), (𝐹‘𝑥)⟩(comp‘𝐷)(𝐾‘𝑥))((𝑦𝐺𝑥)‘𝑚)) = (((𝑦𝐿𝑥)‘𝑚)(⟨(𝐹‘𝑦), (𝐾‘𝑦)⟩(comp‘𝐷)(𝐾‘𝑥))(𝐴‘𝑦)))
31		eqid 2739	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
32	3, 31, 11	funcf1 17824	. . . . . . 7 ⊢ (𝜑 → 𝐾:(Base‘𝐶)⟶(Base‘𝐷))
33	32	ad2antrr 732	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝐾:(Base‘𝐶)⟶(Base‘𝐷))
34	33, 26	ffvelcdmd 7026	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → (𝐾‘𝑥) ∈ (Base‘𝐷))
35	3, 31, 13	funcf1 17824	. . . . . . 7 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
36	35	ad2antrr 732	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
37	36, 26	ffvelcdmd 7026	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
38	36, 25	ffvelcdmd 7026	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → (𝐹‘𝑦) ∈ (Base‘𝐷))
39	31, 24, 8, 34, 37, 38	oppcco 17674	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → (((𝑦𝐺𝑥)‘𝑚)(⟨(𝐾‘𝑥), (𝐹‘𝑥)⟩(comp‘𝑃)(𝐹‘𝑦))(𝐴‘𝑥)) = ((𝐴‘𝑥)(⟨(𝐹‘𝑦), (𝐹‘𝑥)⟩(comp‘𝐷)(𝐾‘𝑥))((𝑦𝐺𝑥)‘𝑚)))
40	33, 25	ffvelcdmd 7026	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → (𝐾‘𝑦) ∈ (Base‘𝐷))
41	31, 24, 8, 34, 40, 38	oppcco 17674	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → ((𝐴‘𝑦)(⟨(𝐾‘𝑥), (𝐾‘𝑦)⟩(comp‘𝑃)(𝐹‘𝑦))((𝑦𝐿𝑥)‘𝑚)) = (((𝑦𝐿𝑥)‘𝑚)(⟨(𝐹‘𝑦), (𝐾‘𝑦)⟩(comp‘𝐷)(𝐾‘𝑥))(𝐴‘𝑦)))
42	30, 39, 41	3eqtr4rd 2785	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → ((𝐴‘𝑦)(⟨(𝐾‘𝑥), (𝐾‘𝑦)⟩(comp‘𝑃)(𝐹‘𝑦))((𝑦𝐿𝑥)‘𝑚)) = (((𝑦𝐺𝑥)‘𝑚)(⟨(𝐾‘𝑥), (𝐹‘𝑥)⟩(comp‘𝑃)(𝐹‘𝑦))(𝐴‘𝑥)))
43		ovtpos 8181	. . . . 5 ⊢ (𝑥tpos 𝐿𝑦) = (𝑦𝐿𝑥)
44	43	fveq1i 6828	. . . 4 ⊢ ((𝑥tpos 𝐿𝑦)‘𝑚) = ((𝑦𝐿𝑥)‘𝑚)
45	44	oveq2i 7367	. . 3 ⊢ ((𝐴‘𝑦)(⟨(𝐾‘𝑥), (𝐾‘𝑦)⟩(comp‘𝑃)(𝐹‘𝑦))((𝑥tpos 𝐿𝑦)‘𝑚)) = ((𝐴‘𝑦)(⟨(𝐾‘𝑥), (𝐾‘𝑦)⟩(comp‘𝑃)(𝐹‘𝑦))((𝑦𝐿𝑥)‘𝑚))
46		ovtpos 8181	. . . . 5 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥)
47	46	fveq1i 6828	. . . 4 ⊢ ((𝑥tpos 𝐺𝑦)‘𝑚) = ((𝑦𝐺𝑥)‘𝑚)
48	47	oveq1i 7366	. . 3 ⊢ (((𝑥tpos 𝐺𝑦)‘𝑚)(⟨(𝐾‘𝑥), (𝐹‘𝑥)⟩(comp‘𝑃)(𝐹‘𝑦))(𝐴‘𝑥)) = (((𝑦𝐺𝑥)‘𝑚)(⟨(𝐾‘𝑥), (𝐹‘𝑥)⟩(comp‘𝑃)(𝐹‘𝑦))(𝐴‘𝑥))
49	42, 45, 48	3eqtr4g 2799	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → ((𝐴‘𝑦)(⟨(𝐾‘𝑥), (𝐾‘𝑦)⟩(comp‘𝑃)(𝐹‘𝑦))((𝑥tpos 𝐿𝑦)‘𝑚)) = (((𝑥tpos 𝐺𝑦)‘𝑚)(⟨(𝐾‘𝑥), (𝐹‘𝑥)⟩(comp‘𝑃)(𝐹‘𝑦))(𝐴‘𝑥)))
50	1, 4, 5, 6, 7, 12, 14, 15, 21, 49	isnatd 49713	1 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐾, tpos 𝐿⟩𝑀⟨𝐹, tpos 𝐺⟩))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ⟨cop 4561  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  tpos ctpos 8165  Basecbs 17170  Hom chom 17222  compcco 17223  oppCatcoppc 17668   Nat cnat 17902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-oppc 17669  df-func 17816  df-nat 17904

This theorem is referenced by:  natoppf2  49720