Theorem 2termoinv 18069

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 2763	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2763	. . . . 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 18060	. . . . . . 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 18060	. . . . . . 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 17736	. . . 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 18054	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (TermO‘𝐶)) → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
18	6, 17	mpdan 699	. . . . . . 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 2849	. . . . 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 4606	. . . . 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 2763	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
25		eqid 2763	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
26	1, 2, 3, 24, 25, 5, 9, 13, 14, 15	issect2 17806	. . 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 17736	. . . 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 18054	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (TermO‘𝐶)) → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
30	10, 29	mpdan 699	. . . . . . 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 2849	. . . . 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 4606	. . . . 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 17806	. . 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 2763	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
39	1, 38, 4, 8, 12, 25	isinv 17812	. . 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 725	1 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {csn 4589  ⟨cop 4595   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  Hom chom 17316  compcco 17317  Catccat 17715  Idccid 17716  Sectcsect 17796  Invcinv 17797  TermOctermo 18034

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

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  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-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-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-id 5556  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-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-1st 7982  df-2nd 7983  df-cat 17719  df-cid 17720  df-sect 17799  df-inv 17800  df-termo 18037

This theorem is referenced by:  termoeu1  18070