Theorem mndtccatid 50500

Description: Lemma for mndtccat 50501 and mndtcid 50502. (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 2763	. 2 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶))
2		eqidd 2763	. 2 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐶))
3		eqidd 2763	. 2 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐶))
4		mndtccat.c	. . 3 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))
5		fvexd 6897	. . 3 ⊢ (𝜑 → (MndToCat‘𝑀) ∈ V)
6	4, 5	eqeltrd 2862	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
7		biid 264	. 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 2762	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
10		eqid 2762	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
11	9, 10	mndidcl 18854	. . . . 5 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
12	8, 11	syl 18	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ (Base‘𝑀))
13	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (0g‘𝑀) ∈ (Base‘𝑀))
14	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 = (MndToCat‘𝑀))
15	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑀 ∈ Mnd)
16		eqidd 2763	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐶))
17		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
18		eqidd 2763	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (Hom ‘𝐶) = (Hom ‘𝐶))
19	14, 15, 16, 17, 17, 18	mndtchom 50497	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑦(Hom ‘𝐶)𝑦) = (Base‘𝑀))
20	13, 19	eleqtrrd 2865	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (0g‘𝑀) ∈ (𝑦(Hom ‘𝐶)𝑦))
21	4	adantr 486	. . . . 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 486	. . . . 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 2763	. . . . 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 1249	. . . . 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 1250	. . . . 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 2763	. . . . 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 50498	. . . 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 7433	. . 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 1282	. . . . 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 2763	. . . . . 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 50497	. . . . 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 2864	. . . 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 2762	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
34	9, 33, 10	mndlid 18859	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑓 ∈ (Base‘𝑀)) → ((0g‘𝑀)(+g‘𝑀)𝑓) = 𝑓)
35	22, 32, 34	syl2anc 596	. . 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 2797	. 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 1251	. . . . 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 50498	. . . 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 7433	. . 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 1283	. . . . 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 50497	. . . . 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 2864	. . . 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 18860	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑔 ∈ (Base‘𝑀)) → (𝑔(+g‘𝑀)(0g‘𝑀)) = 𝑔)
44	22, 42, 43	syl2anc 596	. . 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 2797	. 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 18846	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑔 ∈ (Base‘𝑀) ∧ 𝑓 ∈ (Base‘𝑀)) → (𝑔(+g‘𝑀)𝑓) ∈ (Base‘𝑀))
47	22, 42, 32, 46	syl3anc 1398	. . 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 50498	. . . 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 7433	. . 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 50497	. . 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 2877	. 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 1284	. . . . 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 1252	. . . . . 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 50497	. . . . 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 2864	. . . 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 18847	. . . 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 1399	. . 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 50498	. . . 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 50498	. . . . 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 7433	. . . 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 2763	. . . 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 7437	. . 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 50498	. . . 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 2763	. . . 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 7437	. . 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 2807	. 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 17773	1 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0g‘𝑀))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ⟨cop 4593   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  +_gcplusg 17346  Hom chom 17357  compcco 17358  0_gc0g 17528  Catccat 17756  Idccid 17757  Mndcmnd 18838  MndToCatcmndtc 50490

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-z 12619  df-dec 12740  df-uz 12891  df-fz 13564  df-struct 17243  df-slot 17278  df-ndx 17290  df-base 17306  df-hom 17370  df-cco 17371  df-0g 17530  df-cat 17760  df-cid 17761  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-mndtc 50491

This theorem is used by:  mndtccat  50501  mndtcid  50502