Theorem mndtccatid 50249

Description: Lemma for mndtccat 50250 and mndtcid 50251. (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 2770	. 2 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶))
2		eqidd 2770	. 2 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐶))
3		eqidd 2770	. 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 2869	. 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 2769	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
10		eqid 2769	. . . . . 6 ⊢ (0g‘𝑀) = (0g‘𝑀)
11	9, 10	mndidcl 18806	. . . . 5 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
12	8, 11	syl 18	. . . 4 ⊢ (𝜑 → (0g‘𝑀) ∈ (Base‘𝑀))
13	12	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (0g‘𝑀) ∈ (Base‘𝑀))
14	4	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 = (MndToCat‘𝑀))
15	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑀 ∈ Mnd)
16		eqidd 2770	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐶))
17		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
18		eqidd 2770	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (Hom ‘𝐶) = (Hom ‘𝐶))
19	14, 15, 16, 17, 17, 18	mndtchom 50246	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑦(Hom ‘𝐶)𝑦) = (Base‘𝑀))
20	13, 19	eleqtrrd 2872	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (0g‘𝑀) ∈ (𝑦(Hom ‘𝐶)𝑦))
21	4	adantr 485	. . . . 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 485	. . . . 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 2770	. . . . 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 1247	. . . . 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 1248	. . . . 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 2770	. . . . 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 50247	. . . 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 7428	. . 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 1280	. . . . 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 2770	. . . . . 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 50246	. . . . 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 2871	. . . 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 2769	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
34	9, 33, 10	mndlid 18811	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑓 ∈ (Base‘𝑀)) → ((0g‘𝑀)(+g‘𝑀)𝑓) = 𝑓)
35	22, 32, 34	syl2anc 595	. . 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 2804	. 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 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)))) → 𝑧 ∈ (Base‘𝐶))
38	21, 22, 23, 25, 25, 37, 26	mndtcco 50247	. . . 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 7428	. . 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 1281	. . . . 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 50246	. . . . 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 2871	. . . 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 18812	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑔 ∈ (Base‘𝑀)) → (𝑔(+g‘𝑀)(0g‘𝑀)) = 𝑔)
44	22, 42, 43	syl2anc 595	. . 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 2804	. 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 18799	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑔 ∈ (Base‘𝑀) ∧ 𝑓 ∈ (Base‘𝑀)) → (𝑔(+g‘𝑀)𝑓) ∈ (Base‘𝑀))
47	22, 42, 32, 46	syl3anc 1396	. . 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 50247	. . . 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 7428	. . 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 50246	. . 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 2884	. 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 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)))) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑤))
53		simpr2r 1250	. . . . . 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 50246	. . . . 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 2871	. . . 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 18800	. . . 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 1397	. . 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 50247	. . . 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 50247	. . . . 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 7428	. . . 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 2770	. . . 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 7432	. . 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 50247	. . . 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 2770	. . . 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 7432	. . 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 2814	. 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 17736	1 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0g‘𝑀))))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   ↦ cmpt 5196  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  +_gcplusg 17309  Hom chom 17320  compcco 17321  0_gc0g 17491  Catccat 17719  Idccid 17720  Mndcmnd 18791  MndToCatcmndtc 50239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-struct 17206  df-slot 17241  df-ndx 17253  df-base 17269  df-hom 17333  df-cco 17334  df-0g 17493  df-cat 17723  df-cid 17724  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-mndtc 50240

This theorem is referenced by:  mndtccat  50250  mndtcid  50251