Theorem m2cpminvid2lem 22802

Description: Lemma for m2cpminvid2 22803. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)

Hypotheses

Ref	Expression
m2cpminvid2lem.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
m2cpminvid2lem.p	⊢ 𝑃 = (Poly1‘𝑅)

Assertion

Ref	Expression
m2cpminvid2lem	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))

Distinct variable groups:   𝑛,𝑀   𝑛,𝑁   𝑃,𝑛   𝑅,𝑛   𝑆,𝑛   𝑥,𝑛   𝑦,𝑛

Allowed substitution hints:   𝑃(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑀(𝑥,𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem m2cpminvid2lem

Dummy variables 𝑖 𝑗 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		m2cpminvid2lem.s	. . . . . . . 8 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
2		m2cpminvid2lem.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
3		eqid 2761	. . . . . . . 8 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
4		eqid 2761	. . . . . . . 8 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
5	1, 2, 3, 4	cpmatelimp 22760	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → (𝑀 ∈ (Base‘(𝑁 Mat 𝑃)) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))))
6	5	3impia 1129	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑀 ∈ (Base‘(𝑁 Mat 𝑃)) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅)))
7	6	simprd 499	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))
8	7	adantr 484	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))
9		fvoveq1 7414	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑗)))
10	9	fveq1d 6864	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑘) = ((coe1‘(𝑥𝑀𝑗))‘𝑘))
11	10	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → (((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅)))
12	11	ralbidv 3184	. . . . . . 7 ⊢ (𝑖 = 𝑥 → (∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅)))
13		oveq2 7399	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑥𝑀𝑗) = (𝑥𝑀𝑦))
14	13	fveq2d 6866	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (coe1‘(𝑥𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
15	14	fveq1d 6864	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((coe1‘(𝑥𝑀𝑗))‘𝑘) = ((coe1‘(𝑥𝑀𝑦))‘𝑘))
16	15	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅)))
17	16	ralbidv 3184	. . . . . . 7 ⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅)))
18	12, 17	rspc2v 3591	. . . . . 6 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅) → ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅)))
19	18	adantl 485	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅) → ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅)))
20		fveqeq2 6871	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅) ↔ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅)))
21	20	cbvralvw 3239	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅) ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅))
22		simpl2 1205	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑅 ∈ Ring)
23		eqid 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘𝑃)
24		simprl 780	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
25		simprr 782	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
26	1, 2, 3, 4	cpmatpmat 22758	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ (Base‘(𝑁 Mat 𝑃)))
27	26	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑀 ∈ (Base‘(𝑁 Mat 𝑃)))
28	3, 23, 4, 24, 25, 27	matecld 22474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘𝑃))
29		0nn0 12490	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ0
30		eqid 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
31		eqid 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	30, 23, 2, 31	coe1fvalcl 22262	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥𝑀𝑦) ∈ (Base‘𝑃) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
33	28, 29, 32	sylancl 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
34	22, 33	jca 519	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
35	34	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → (𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
36		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
37		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) = (0g‘𝑅)
38	2, 36, 31, 37	coe1scl 22338	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → (coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅))))
39	35, 38	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → (coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅))))
40	39	fveq1d 6864	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)))‘𝑛))
41		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅))))
42		eqeq1 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
43	42	ifbid 4501	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝑛 → if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)) = if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)))
44	43	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)) = if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)))
45		nnnn0 12482	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
46	45	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
47		fvex 6875	. . . . . . . . . . . . . . . 16 ⊢ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ V
48		fvex 6875	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) ∈ V
49	47, 48	ifex 4528	. . . . . . . . . . . . . . 15 ⊢ if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)) ∈ V
50	49	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)) ∈ V)
51	41, 44, 46, 50	fvmptd 6978	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)))‘𝑛) = if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)))
52		nnne0 12241	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
53	52	neneqd 2961	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
54	53	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
55	54	iffalsed 4488	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)) = (0g‘𝑅))
56	40, 51, 55	3eqtrd 2800	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = (0g‘𝑅))
57		eqcom 2768	. . . . . . . . . . . . 13 ⊢ (((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅) ↔ (0g‘𝑅) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
58	57	biimpi 218	. . . . . . . . . . . 12 ⊢ (((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅) → (0g‘𝑅) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
59	56, 58	sylan9eq 2816	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅)) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
60	59	ex 416	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → (((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛)))
61	60	ralimdva 3173	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅) → ∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛)))
62	61	imp 410	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅)) → ∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
63	34	adantr 484	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅)) → (𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
64	2, 36, 31	ply1sclid 22339	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0))
65	64	eqcomd 2767	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
66	63, 65	syl 17	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅)) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
67	62, 66	jca 519	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅)) → (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0)))
68	67	ex 416	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅) → (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))))
69	21, 68	biimtrid 244	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅) → (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))))
70	19, 69	syld 47	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅) → (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))))
71	8, 70	mpd 15	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0)))
72		c0ex 11167	. . . 4 ⊢ 0 ∈ V
73		fveq2 6862	. . . . . 6 ⊢ (𝑛 = 0 → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0))
74		fveq2 6862	. . . . . 6 ⊢ (𝑛 = 0 → ((coe1‘(𝑥𝑀𝑦))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘0))
75	73, 74	eqeq12d 2777	. . . . 5 ⊢ (𝑛 = 0 → (((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ↔ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0)))
76	75	ralunsn 4849	. . . 4 ⊢ (0 ∈ V → (∀𝑛 ∈ (ℕ ∪ {0})((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ↔ (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))))
77	72, 76	mp1i 13	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (∀𝑛 ∈ (ℕ ∪ {0})((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ↔ (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))))
78	71, 77	mpbird 259	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ∀𝑛 ∈ (ℕ ∪ {0})((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
79		df-n0 12476	. . 3 ⊢ ℕ0 = (ℕ ∪ {0})
80	79	raleqi 3317	. 2 ⊢ (∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ↔ ∀𝑛 ∈ (ℕ ∪ {0})((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
81	78, 80	sylibr 236	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ∪ cun 3900  ifcif 4477  {csn 4579   ↦ cmpt 5178  ‘cfv 6516  (class class class)co 7391  Fincfn 8921  0cc0 11067  ℕcn 12204  ℕ₀cn0 12475  Basecbs 17236  0_gc0g 17459  Ringcrg 20270  algSccascl 21892  Poly₁cpl1 22227  coe₁cco1 22228   Mat cmat 22455   ConstPolyMat ccpmat 22751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  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 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-isom 6525  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-of 7655  df-ofr 7656  df-om 7842  df-1st 7965  df-2nd 7966  df-supp 8135  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-2o 8432  df-er 8672  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9302  df-sup 9382  df-oi 9452  df-card 9891  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12476  df-z 12563  df-dec 12683  df-uz 12834  df-fz 13507  df-fzo 13654  df-seq 14009  df-hash 14338  df-struct 17174  df-sets 17191  df-slot 17209  df-ndx 17221  df-base 17237  df-ress 17258  df-plusg 17290  df-mulr 17291  df-sca 17293  df-vsca 17294  df-ip 17295  df-tset 17296  df-ple 17297  df-ds 17299  df-hom 17301  df-cco 17302  df-0g 17461  df-gsum 17462  df-prds 17467  df-pws 17469  df-mre 17605  df-mrc 17606  df-acs 17608  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18969  df-minusg 18970  df-sbg 18971  df-mulg 19101  df-subg 19156  df-ghm 19245  df-cntz 19348  df-cmn 19813  df-abl 19814  df-mgp 20178  df-rng 20190  df-ur 20219  df-ring 20272  df-subrng 20583  df-subrg 20607  df-lmod 20917  df-lss 20987  df-sra 21228  df-rgmod 21229  df-dsmm 21772  df-frlm 21787  df-ascl 21895  df-psr 21949  df-mvr 21950  df-mpl 21951  df-opsr 21953  df-psr1 22230  df-vr1 22231  df-ply1 22232  df-coe1 22233  df-mat 22456  df-cpmat 22754

This theorem is referenced by:  m2cpminvid2  22803