Theorem natoppf 50007

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 17769	. 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 50003	. . 3 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿)
12	2, 8, 11	funcoppc 17927	. 2 ⊢ (𝜑 → 𝐾(𝑂 Func 𝑃)tpos 𝐿)
13	9, 10	natrcl2 50002	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
14	2, 8, 13	funcoppc 17927	. 2 ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)
15	9, 10, 3	natfn 18009	. 2 ⊢ (𝜑 → 𝐴 Fn (Base‘𝐶))
16	10	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))
17		eqid 2763	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
18		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
19	9, 16, 3, 17, 18	natcl 18008	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥)))
20	17, 8	oppchom 17766	. . 3 ⊢ ((𝐾‘𝑥)(Hom ‘𝑃)(𝐹‘𝑥)) = ((𝐹‘𝑥)(Hom ‘𝐷)(𝐾‘𝑥))
21	19, 20	eleqtrrdi 2874	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ ((𝐾‘𝑥)(Hom ‘𝑃)(𝐹‘𝑥)))
22	10	ad2antrr 738	. . . . 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 789	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑦 ∈ (Base‘𝐶))
26		simplrl 788	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑥 ∈ (Base‘𝐶))
27		simpr 489	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦))
28	23, 2	oppchom 17766	. . . . . 6 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
29	27, 28	eleqtrdi 2873	. . . . 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 18010	. . . 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 17918	. . . . . . 7 ⊢ (𝜑 → 𝐾:(Base‘𝐶)⟶(Base‘𝐷))
33	32	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝐾:(Base‘𝐶)⟶(Base‘𝐷))
34	33, 26	ffvelcdmd 7080	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → (𝐾‘𝑥) ∈ (Base‘𝐷))
35	3, 31, 13	funcf1 17918	. . . . . . 7 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
36	35	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
37	36, 26	ffvelcdmd 7080	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑚 ∈ (𝑥(Hom ‘𝑂)𝑦)) → (𝐹‘𝑥) ∈ (Base‘𝐷))
38	36, 25	ffvelcdmd 7080	. . . . 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 17768	. . . 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 7080	. . . . 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 17768	. . . 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 8233	. . . . 5 ⊢ (𝑥tpos 𝐿𝑦) = (𝑦𝐿𝑥)
44	43	fveq1i 6882	. . . 4 ⊢ ((𝑥tpos 𝐿𝑦)‘𝑚) = ((𝑦𝐿𝑥)‘𝑚)
45	44	oveq2i 7421	. . 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 8233	. . . . 5 ⊢ (𝑥tpos 𝐺𝑦) = (𝑦𝐺𝑥)
47	46	fveq1i 6882	. . . 4 ⊢ ((𝑥tpos 𝐺𝑦)‘𝑚) = ((𝑦𝐺𝑥)‘𝑚)
48	47	oveq1i 7420	. . 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 50001	1 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐾, tpos 𝐿⟩𝑀⟨𝐹, tpos 𝐺⟩))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ⟨cop 4595  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  tpos ctpos 8217  Basecbs 17264  Hom chom 17316  compcco 17317  oppCatcoppc 17762   Nat cnat 17996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-hom 17329  df-cco 17330  df-cat 17719  df-cid 17720  df-oppc 17763  df-func 17910  df-nat 17998

This theorem is referenced by:  natoppf2  50008