Theorem natoppf 49200

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 2730	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4	2, 3	oppcbas 17685	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
5		eqid 2730	. 2 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
6		eqid 2730	. 2 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
7		eqid 2730	. 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 49196	. . 3 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)
12	2, 8, 11	funcoppc 17843	. 2 ⊢ (𝜑 → 𝐾(𝑂 Func 𝑃)tpos 𝐿)
13	9, 10	natrcl2 49195	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
14	2, 8, 13	funcoppc 17843	. 2 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)
15	9, 10, 3	natfn 17925	. 2 ⊢ (𝜑 → 𝐴 Fn (Base‘𝐶))
16	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))
17		eqid 2730	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
19	9, 16, 3, 17, 18	natcl 17924	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥)))
20	17, 8	oppchom 17682	. . 3 ⊢ ((𝐾‘𝑥)(Hom ‘𝑃)(𝐹‘𝑥)) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥))
21	19, 20	eleqtrrdi 2840	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ ((𝐾‘𝑥)(Hom ‘𝑃)(𝐹‘𝑥)))
22	10	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))
23		eqid 2730	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
24		eqid 2730	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
25		simplrr 777	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑦 ∈ (Base‘𝐶))
26		simplrl 776	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑥 ∈ (Base‘𝐶))
27		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦))
28	23, 2	oppchom 17682	. . . . . 6 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
29	27, 28	eleqtrdi 2839	. . . . 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 17926	. . . 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 2730	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
32	3, 31, 11	funcf1 17834	. . . . . . 7 ⊢ (𝜑 → 𝐾:(Base‘𝐶)⟶(Base‘𝐷))
33	32	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝐾:(Base‘𝐶)⟶(Base‘𝐷))
34	33, 26	ffvelcdmd 7059	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → (𝐾‘𝑥) ∈ (Base‘𝐷))
35	3, 31, 13	funcf1 17834	. . . . . . 7 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
36	35	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
37	36, 26	ffvelcdmd 7059	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
38	36, 25	ffvelcdmd 7059	. . . . 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 17684	. . . 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 7059	. . . . 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 17684	. . . 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 2776	. . 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 8222	. . . . 5 ⊢ (𝑥tpos 𝐿𝑦) = (𝑦𝐿𝑥)
44	43	fveq1i 6861	. . . 4 ⊢ ((𝑥tpos 𝐿𝑦)‘𝑚) = ((𝑦𝐿𝑥)‘𝑚)
45	44	oveq2i 7400	. . 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 8222	. . . . 5 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥)
47	46	fveq1i 6861	. . . 4 ⊢ ((𝑥tpos 𝐺𝑦)‘𝑚) = ((𝑦𝐺𝑥)‘𝑚)
48	47	oveq1i 7399	. . 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 2790	. 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 49194	1 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐾, tpos 𝐿⟩𝑀⟨𝐹, tpos 𝐺⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4597  ⟶wf 6509  ‘cfv 6513  (class class class)co 7389  tpos ctpos 8206  Basecbs 17185  Hom chom 17237  compcco 17238  oppCatcoppc 17678   Nat cnat 17912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-er 8673  df-map 8803  df-ixp 8873  df-en 8921  df-dom 8922  df-sdom 8923  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-3 12251  df-4 12252  df-5 12253  df-6 12254  df-7 12255  df-8 12256  df-9 12257  df-n0 12449  df-z 12536  df-dec 12656  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-cat 17635  df-cid 17636  df-oppc 17679  df-func 17826  df-nat 17914

This theorem is referenced by:  natoppf2  49201