Theorem mndtccatid 50091

Description: Lemma for mndtccat 50092 and mndtcid 50093. (Contributed by Zhi Wang, 22-Sep-2024.)

Hypotheses

Ref	Expression
mndtccat.c	⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))
mndtccat.m	⊢ (𝜑 → 𝑀 ∈ Mnd)

Assertion

Ref	Expression
mndtccatid	⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0g‘𝑀))))

Distinct variable groups:   𝑦,𝐶   𝜑,𝑦

Allowed substitution hint:   𝑀(𝑦)

Proof of Theorem mndtccatid

Dummy variables 𝑓 𝑔 𝑘 𝑤 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2742	. 2 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶))
2		eqidd 2742	. 2 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐶))
3		eqidd 2742	. 2 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐶))
4		mndtccat.c	. . 3 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))
5		fvexd 6846	. . 3 ⊢ (𝜑 → (MndToCat‘𝑀) ∈ V)
6	4, 5	eqeltrd 2841	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
7		biid 263	. 2 ⊢ (((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤))) ↔ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤))))
8		mndtccat.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Mnd)
9		eqid 2741	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
10		eqid 2741	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
11	9, 10	mndidcl 18712	. . . . 5 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
12	8, 11	syl 17	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ (Base‘𝑀))
13	12	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (0g‘𝑀) ∈ (Base‘𝑀))
14	4	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 = (MndToCat‘𝑀))
15	8	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑀 ∈ Mnd)
16		eqidd 2742	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐶))
17		simpr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
18		eqidd 2742	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (Hom ‘𝐶) = (Hom ‘𝐶))
19	14, 15, 16, 17, 17, 18	mndtchom 50088	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑦(Hom ‘𝐶)𝑦) = (Base‘𝑀))
20	13, 19	eleqtrrd 2844	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (0g‘𝑀) ∈ (𝑦(Hom ‘𝐶)𝑦))
21	4	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝐶 = (MndToCat‘𝑀))
22	8	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑀 ∈ Mnd)
23		eqidd 2742	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (Base‘𝐶) = (Base‘𝐶))
24		simpr1l 1238	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑥 ∈ (Base‘𝐶))
25		simpr1r 1239	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑦 ∈ (Base‘𝐶))
26		eqidd 2742	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (comp‘𝐶) = (comp‘𝐶))
27	21, 22, 23, 24, 25, 25, 26	mndtcco 50089	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑦) = (+g‘𝑀))
28	27	oveqd 7377	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → ((0g‘𝑀)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑦)𝑓) = ((0g‘𝑀)(+g‘𝑀)𝑓))
29		simpr31 1271	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
30		eqidd 2742	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (Hom ‘𝐶) = (Hom ‘𝐶))
31	21, 22, 23, 24, 25, 30	mndtchom 50088	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑥(Hom ‘𝐶)𝑦) = (Base‘𝑀))
32	29, 31	eleqtrd 2843	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑓 ∈ (Base‘𝑀))
33		eqid 2741	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
34	9, 33, 10	mndlid 18717	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑓 ∈ (Base‘𝑀)) → ((0g‘𝑀)(+g‘𝑀)𝑓) = 𝑓)
35	22, 32, 34	syl2anc 591	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → ((0g‘𝑀)(+g‘𝑀)𝑓) = 𝑓)
36	28, 35	eqtrd 2776	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → ((0g‘𝑀)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑦)𝑓) = 𝑓)
37		simpr2l 1240	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑧 ∈ (Base‘𝐶))
38	21, 22, 23, 25, 25, 37, 26	mndtcco 50089	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (⟨𝑦, 𝑦⟩(comp‘𝐶)𝑧) = (+g‘𝑀))
39	38	oveqd 7377	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑔(⟨𝑦, 𝑦⟩(comp‘𝐶)𝑧)(0g‘𝑀)) = (𝑔(+g‘𝑀)(0g‘𝑀)))
40		simpr32 1272	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
41	21, 22, 23, 25, 37, 30	mndtchom 50088	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑦(Hom ‘𝐶)𝑧) = (Base‘𝑀))
42	40, 41	eleqtrd 2843	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑔 ∈ (Base‘𝑀))
43	9, 33, 10	mndrid 18718	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑔 ∈ (Base‘𝑀)) → (𝑔(+g‘𝑀)(0g‘𝑀)) = 𝑔)
44	22, 42, 43	syl2anc 591	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑔(+g‘𝑀)(0g‘𝑀)) = 𝑔)
45	39, 44	eqtrd 2776	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑔(⟨𝑦, 𝑦⟩(comp‘𝐶)𝑧)(0g‘𝑀)) = 𝑔)
46	9, 33	mndcl 18705	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑔 ∈ (Base‘𝑀) ∧ 𝑓 ∈ (Base‘𝑀)) → (𝑔(+g‘𝑀)𝑓) ∈ (Base‘𝑀))
47	22, 42, 32, 46	syl3anc 1380	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑔(+g‘𝑀)𝑓) ∈ (Base‘𝑀))
48	21, 22, 23, 24, 25, 37, 26	mndtcco 50089	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (+g‘𝑀))
49	48	oveqd 7377	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(+g‘𝑀)𝑓))
50	21, 22, 23, 24, 37, 30	mndtchom 50088	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑥(Hom ‘𝐶)𝑧) = (Base‘𝑀))
51	47, 49, 50	3eltr4d 2856	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
52		simpr33 1273	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤))
53		simpr2r 1241	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑤 ∈ (Base‘𝐶))
54	21, 22, 23, 37, 53, 30	mndtchom 50088	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑧(Hom ‘𝐶)𝑤) = (Base‘𝑀))
55	52, 54	eleqtrd 2843	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑘 ∈ (Base‘𝑀))
56	9, 33	mndass 18706	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ (𝑘 ∈ (Base‘𝑀) ∧ 𝑔 ∈ (Base‘𝑀) ∧ 𝑓 ∈ (Base‘𝑀))) → ((𝑘(+g‘𝑀)𝑔)(+g‘𝑀)𝑓) = (𝑘(+g‘𝑀)(𝑔(+g‘𝑀)𝑓)))
57	22, 55, 42, 32, 56	syl13anc 1381	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → ((𝑘(+g‘𝑀)𝑔)(+g‘𝑀)𝑓) = (𝑘(+g‘𝑀)(𝑔(+g‘𝑀)𝑓)))
58	21, 22, 23, 24, 25, 53, 26	mndtcco 50089	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤) = (+g‘𝑀))
59	21, 22, 23, 25, 37, 53, 26	mndtcco 50089	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤) = (+g‘𝑀))
60	59	oveqd 7377	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) = (𝑘(+g‘𝑀)𝑔))
61		eqidd 2742	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑓 = 𝑓)
62	58, 60, 61	oveq123d 7381	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → ((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = ((𝑘(+g‘𝑀)𝑔)(+g‘𝑀)𝑓))
63	21, 22, 23, 24, 37, 53, 26	mndtcco 50089	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤) = (+g‘𝑀))
64		eqidd 2742	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → 𝑘 = 𝑘)
65	63, 64, 49	oveq123d 7381	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (𝑘(+g‘𝑀)(𝑔(+g‘𝑀)𝑓)))
66	57, 62, 65	3eqtr4d 2786	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ (𝑧 ∈ (Base‘𝐶) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤)))) → ((𝑘(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
67	1, 2, 3, 6, 7, 20, 36, 45, 51, 66	iscatd2 17642	1 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0g‘𝑀))))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  Vcvv 3433  ⟨cop 4564   ↦ cmpt 5156  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  +_gcplusg 17215  Hom chom 17226  compcco 17227  0_gc0g 17397  Catccat 17625  Idccid 17626  Mndcmnd 18697  MndToCatcmndtc 50081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-struct 17112  df-slot 17147  df-ndx 17159  df-base 17175  df-hom 17239  df-cco 17240  df-0g 17399  df-cat 17629  df-cid 17630  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mndtc 50082

This theorem is referenced by:  mndtccat  50092  mndtcid  50093