Theorem 2termoinv 18096

Description: Morphisms between two terminal objects are inverses. (Contributed by AV, 18-Apr-2020.)

Hypotheses

Ref	Expression
termoeu1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
termoeu1.a	⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶))
termoeu1.b	⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶))

Assertion

Ref	Expression
2termoinv	⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)

Proof of Theorem 2termoinv

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2765	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2765	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		termoeu1.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	4	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐶 ∈ Cat)
6		termoeu1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶))
7		termoo 18087	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
8	4, 6, 7	sylc 66	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶))
9	8	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐴 ∈ (Base‘𝐶))
10		termoeu1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶))
11		termoo 18087	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (𝐵 ∈ (TermO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
12	4, 10, 11	sylc 66	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (Base‘𝐶))
13	12	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐵 ∈ (Base‘𝐶))
14		simp3 1156	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵))
15		simp2 1155	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴))
16	1, 2, 3, 5, 9, 13, 9, 14, 15	catcocl 17763	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ (𝐴(Hom ‘𝐶)𝐴))
17	1, 2, 4	termoid 18081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (TermO‘𝐶)) → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
18	6, 17	mpdan 700	. . . . . . 7 ⊢ (𝜑 → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
19	18	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
20	19	eleq2d 2851	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ (𝐴(Hom ‘𝐶)𝐴) ↔ (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ {((Id‘𝐶)‘𝐴)}))
21		elsni 4608	. . . . 5 ⊢ ((𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ {((Id‘𝐶)‘𝐴)} → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴))
22	20, 21	biimtrdi 256	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ (𝐴(Hom ‘𝐶)𝐴) → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴)))
23	16, 22	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴))
24		eqid 2765	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
25		eqid 2765	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
26	1, 2, 3, 24, 25, 5, 9, 13, 14, 15	issect2 17833	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(𝐴(Sect‘𝐶)𝐵)𝐺 ↔ (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴)))
27	23, 26	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Sect‘𝐶)𝐵)𝐺)
28	1, 2, 3, 5, 13, 9, 13, 15, 14	catcocl 17763	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ (𝐵(Hom ‘𝐶)𝐵))
29	1, 2, 4	termoid 18081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (TermO‘𝐶)) → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
30	10, 29	mpdan 700	. . . . . . 7 ⊢ (𝜑 → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
31	30	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
32	31	eleq2d 2851	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ (𝐵(Hom ‘𝐶)𝐵) ↔ (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ {((Id‘𝐶)‘𝐵)}))
33		elsni 4608	. . . . 5 ⊢ ((𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ {((Id‘𝐶)‘𝐵)} → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵))
34	32, 33	biimtrdi 256	. . . 4 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ((𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ (𝐵(Hom ‘𝐶)𝐵) → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵)))
35	28, 34	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵))
36	1, 2, 3, 24, 25, 5, 13, 9, 15, 14	issect2 17833	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐺(𝐵(Sect‘𝐶)𝐴)𝐹 ↔ (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵)))
37	35, 36	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐺(𝐵(Sect‘𝐶)𝐴)𝐹)
38		eqid 2765	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
39	1, 38, 4, 8, 12, 25	isinv 17839	. . 3 ⊢ (𝜑 → (𝐹(𝐴(Inv‘𝐶)𝐵)𝐺 ↔ (𝐹(𝐴(Sect‘𝐶)𝐵)𝐺 ∧ 𝐺(𝐵(Sect‘𝐶)𝐴)𝐹)))
40	39	3ad2ant1 1151	. 2 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐹(𝐴(Inv‘𝐶)𝐵)𝐺 ↔ (𝐹(𝐴(Sect‘𝐶)𝐵)𝐺 ∧ 𝐺(𝐵(Sect‘𝐶)𝐴)𝐹)))
41	27, 37, 40	mpbir2and 726	1 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {csn 4591  ⟨cop 4597   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  Hom chom 17343  compcco 17344  Catccat 17742  Idccid 17743  Sectcsect 17823  Invcinv 17824  TermOctermo 18061

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-cat 17746  df-cid 17747  df-sect 17826  df-inv 17827  df-termo 18064

This theorem is used by:  termoeu1  18097