Theorem natoppf 50682

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 2763	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4	2, 3	oppcbas 17942	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
5		eqid 2763	. 2 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
6		eqid 2763	. 2 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
7		eqid 2763	. 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 50672	. . 3 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)
12	2, 8, 11	funcoppc 18100	. 2 ⊢ (𝜑 → 𝐾(𝑂 Func 𝑃)tpos 𝐿)
13	9, 10	natrcl2 50671	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
14	2, 8, 13	funcoppc 18100	. 2 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)
15	9, 10, 3	natfn 18182	. 2 ⊢ (𝜑 → 𝐴 Fn (Base‘𝐶))
16	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))
17		eqid 2763	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
19	9, 16, 3, 17, 18	natcl 18181	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥)))
20	17, 8	oppchom 17939	. . 3 ⊢ ((𝐾‘𝑥)(Hom ‘𝑃)(𝐹‘𝑥)) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥))
21	19, 20	eleqtrrdi 2875	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ ((𝐾‘𝑥)(Hom ‘𝑃)(𝐹‘𝑥)))
22	10	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))
23		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
24		eqid 2763	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
25		simplrr 790	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑦 ∈ (Base‘𝐶))
26		simplrl 789	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑥 ∈ (Base‘𝐶))
27		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦))
28	23, 2	oppchom 17939	. . . . . 6 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
29	27, 28	eleqtrdi 2874	. . . . 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 18183	. . . 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 2763	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
32	3, 31, 11	funcf1 18091	. . . . . . 7 ⊢ (𝜑 → 𝐾:(Base‘𝐶)⟶(Base‘𝐷))
33	32	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝐾:(Base‘𝐶)⟶(Base‘𝐷))
34	33, 26	ffvelcdmd 7117	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → (𝐾‘𝑥) ∈ (Base‘𝐷))
35	3, 31, 13	funcf1 18091	. . . . . . 7 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
36	35	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
37	36, 26	ffvelcdmd 7117	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
38	36, 25	ffvelcdmd 7117	. . . . 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 17941	. . . 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 7117	. . . . 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 17941	. . . 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 2809	. . 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 8296	. . . . 5 ⊢ (𝑥tpos 𝐿𝑦) = (𝑦𝐿𝑥)
44	43	fveq1i 6917	. . . 4 ⊢ ((𝑥tpos 𝐿𝑦)‘𝑚) = ((𝑦𝐿𝑥)‘𝑚)
45	44	oveq2i 7462	. . 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 8296	. . . . 5 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥)
47	46	fveq1i 6917	. . . 4 ⊢ ((𝑥tpos 𝐺𝑦)‘𝑚) = ((𝑦𝐺𝑥)‘𝑚)
48	47	oveq1i 7461	. . 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 2823	. 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 50670	1 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐾, tpos 𝐿⟩𝑀⟨𝐹, tpos 𝐺⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ⟨cop 4840  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  tpos ctpos 8280  Basecbs 17431  Hom chom 17483  compcco 17484  oppCatcoppc 17935   Nat cnat 18169

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-map 8887  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-hom 17497  df-cco 17498  df-cat 17892  df-cid 17893  df-oppc 17936  df-func 18083  df-nat 18171

This theorem is used by:  natoppf2  50683