2termoinv - Metamath Proof Explorer

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Theorem 2termoinv 17966

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 2732	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2732	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2732	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		termoeu1.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	4	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐶 ∈ Cat)
6		termoeu1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶))
7		termoo 17957	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶)))
8	4, 6, 7	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶))
9	8	3ad2ant1 1133	. . . . 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 17957	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (𝐵 ∈ (TermO‘𝐶) → 𝐵 ∈ (Base‘𝐶)))
12	4, 10, 11	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (Base‘𝐶))
13	12	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐵 ∈ (Base‘𝐶))
14		simp3 1138	. . . . 5 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵))
15		simp2 1137	. . . . 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 17628	. . . 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 17951	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (TermO‘𝐶)) → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
18	6, 17	mpdan 685	. . . . . . 7 ⊢ (𝜑 → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
19	18	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐴(Hom ‘𝐶)𝐴) = {((Id‘𝐶)‘𝐴)})
20	19	eleq2d 2819	. . . . 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 4645	. . . . 5 ⊢ ((𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) ∈ {((Id‘𝐶)‘𝐴)} → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴))
22	20, 21	syl6bi 252	. . . 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 15	. . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐺(⟨𝐴, 𝐵⟩(comp‘𝐶)𝐴)𝐹) = ((Id‘𝐶)‘𝐴))
24		eqid 2732	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
25		eqid 2732	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
26	1, 2, 3, 24, 25, 5, 9, 13, 14, 15	issect2 17700	. . 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 256	. 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 17628	. . . 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 17951	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (TermO‘𝐶)) → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
30	10, 29	mpdan 685	. . . . . . 7 ⊢ (𝜑 → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
31	30	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → (𝐵(Hom ‘𝐶)𝐵) = {((Id‘𝐶)‘𝐵)})
32	31	eleq2d 2819	. . . . 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 4645	. . . . 5 ⊢ ((𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) ∈ {((Id‘𝐶)‘𝐵)} → (𝐹(⟨𝐵, 𝐴⟩(comp‘𝐶)𝐵)𝐺) = ((Id‘𝐶)‘𝐵))
34	32, 33	syl6bi 252	. . . 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 15	. . 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 17700	. . 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 256	. 2 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐺(𝐵(Sect‘𝐶)𝐴)𝐹)
38		eqid 2732	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
39	1, 38, 4, 8, 12, 25	isinv 17706	. . 3 ⊢ (𝜑 → (𝐹(𝐴(Inv‘𝐶)𝐵)𝐺 ↔ (𝐹(𝐴(Sect‘𝐶)𝐵)𝐺 ∧ 𝐺(𝐵(Sect‘𝐶)𝐴)𝐹)))
40	39	3ad2ant1 1133	. 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 711	1 ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {csn 4628 ⟨cop 4634 class class class wbr 5148 ‘cfv 6543 (class class class)co 7408 Basecbs 17143 Hom chom 17207 compcco 17208 Catccat 17607 Idccid 17608 Sectcsect 17690 Invcinv 17691 TermOctermo 17931

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-cat 17611 df-cid 17612 df-sect 17693 df-inv 17694 df-termo 17934

This theorem is referenced by: termoeu1 17967

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