mndtccatid - Mathbox for Zhi Wang

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Theorem mndtccatid 48211

Description: Lemma for mndtccat 48212 and mndtcid 48213. (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) ↦ (0_g‘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 2726	. 2 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶))
2		eqidd 2726	. 2 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐶))
3		eqidd 2726	. 2 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐶))
4		mndtccat.c	. . 3 ⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))
5		fvexd 6907	. . 3 ⊢ (𝜑 → (MndToCat‘𝑀) ∈ V)
6	4, 5	eqeltrd 2825	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
7		biid 260	. 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 2725	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
10		eqid 2725	. . . . . 6 ⊢ (0_g‘𝑀) = (0_g‘𝑀)
11	9, 10	mndidcl 18708	. . . . 5 ⊢ (𝑀 ∈ Mnd → (0_g‘𝑀) ∈ (Base‘𝑀))
12	8, 11	syl 17	. . . 4 ⊢ (𝜑 → (0_g‘𝑀) ∈ (Base‘𝑀))
13	12	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (0_g‘𝑀) ∈ (Base‘𝑀))
14	4	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐶 = (MndToCat‘𝑀))
15	8	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑀 ∈ Mnd)
16		eqidd 2726	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐶))
17		simpr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
18		eqidd 2726	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (Hom ‘𝐶) = (Hom ‘𝐶))
19	14, 15, 16, 17, 17, 18	mndtchom 48208	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑦(Hom ‘𝐶)𝑦) = (Base‘𝑀))
20	13, 19	eleqtrrd 2828	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → (0_g‘𝑀) ∈ (𝑦(Hom ‘𝐶)𝑦))
21	4	adantr 479	. . . . 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 479	. . . . 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 2726	. . . . 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 1227	. . . . 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 1228	. . . . 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 2726	. . . . 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 48209	. . . 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)))) → ((0_g‘𝑀)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑦)𝑓) = ((0_g‘𝑀)(+_g‘𝑀)𝑓))
29		simpr31 1260	. . . . 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 2726	. . . . . 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 48208	. . . . 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 2827	. . . 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 2725	. . . . 5 ⊢ (+_g‘𝑀) = (+_g‘𝑀)
34	9, 33, 10	mndlid 18713	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑓 ∈ (Base‘𝑀)) → ((0_g‘𝑀)(+_g‘𝑀)𝑓) = 𝑓)
35	22, 32, 34	syl2anc 582	. . 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)))) → ((0_g‘𝑀)(+_g‘𝑀)𝑓) = 𝑓)
36	28, 35	eqtrd 2765	. 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)))) → ((0_g‘𝑀)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑦)𝑓) = 𝑓)
37		simpr2l 1229	. . . . 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 48209	. . . 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)(0_g‘𝑀)) = (𝑔(+_g‘𝑀)(0_g‘𝑀)))
40		simpr32 1261	. . . . 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 48208	. . . . 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 2827	. . . 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 18714	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑔 ∈ (Base‘𝑀)) → (𝑔(+_g‘𝑀)(0_g‘𝑀)) = 𝑔)
44	22, 42, 43	syl2anc 582	. . 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)(0_g‘𝑀)) = 𝑔)
45	39, 44	eqtrd 2765	. 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)(0_g‘𝑀)) = 𝑔)
46	9, 33	mndcl 18701	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑔 ∈ (Base‘𝑀) ∧ 𝑓 ∈ (Base‘𝑀)) → (𝑔(+_g‘𝑀)𝑓) ∈ (Base‘𝑀))
47	22, 42, 32, 46	syl3anc 1368	. . 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 48209	. . . 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 48208	. . 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 2840	. 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 1262	. . . . 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 1230	. . . . . 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 48208	. . . . 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 2827	. . . 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 18702	. . . 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 1369	. . 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 48209	. . . 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 48209	. . . . 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 2726	. . . 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 48209	. . . 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 2726	. . . 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 2775	. 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 17660	1 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0_g‘𝑀))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⟨cop 4630 ↦ cmpt 5226 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 +_gcplusg 17232 Hom chom 17243 compcco 17244 0_gc0g 17420 Catccat 17643 Idccid 17644 Mndcmnd 18693 MndToCatcmndtc 48201

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-hom 17256 df-cco 17257 df-0g 17422 df-cat 17647 df-cid 17648 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mndtc 48202

This theorem is referenced by: mndtccat 48212 mndtcid 48213

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