Theorem m2cpminvid2lem 22715

Description: Lemma for m2cpminvid2 22716. (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 2737	. . . . . . . 8 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
4		eqid 2737	. . . . . . . 8 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
5	1, 2, 3, 4	cpmatelimp 22673	. . . . . . 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 1118	. . . . . 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 495	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))
8	7	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))
9		fvoveq1 7393	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑗)))
10	9	fveq1d 6846	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑘) = ((coe1‘(𝑥𝑀𝑗))‘𝑘))
11	10	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → (((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅)))
12	11	ralbidv 3161	. . . . . . 7 ⊢ (𝑖 = 𝑥 → (∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅)))
13		oveq2 7378	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑥𝑀𝑗) = (𝑥𝑀𝑦))
14	13	fveq2d 6848	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (coe1‘(𝑥𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
15	14	fveq1d 6846	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((coe1‘(𝑥𝑀𝑗))‘𝑘) = ((coe1‘(𝑥𝑀𝑦))‘𝑘))
16	15	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅)))
17	16	ralbidv 3161	. . . . . . 7 ⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅)))
18	12, 17	rspc2v 3589	. . . . . 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 481	. . . . 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 6853	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅) ↔ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅)))
21	20	cbvralvw 3216	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅) ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅))
22		simpl2 1194	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑅 ∈ Ring)
23		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘𝑃)
24		simprl 771	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
25		simprr 773	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
26	1, 2, 3, 4	cpmatpmat 22671	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ (Base‘(𝑁 Mat 𝑃)))
27	26	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑀 ∈ (Base‘(𝑁 Mat 𝑃)))
28	3, 23, 4, 24, 25, 27	matecld 22387	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘𝑃))
29		0nn0 12430	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ0
30		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
31		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	30, 23, 2, 31	coe1fvalcl 22170	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥𝑀𝑦) ∈ (Base‘𝑃) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
33	28, 29, 32	sylancl 587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
34	22, 33	jca 511	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
35	34	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → (𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
36		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
37		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) = (0g‘𝑅)
38	2, 36, 31, 37	coe1scl 22246	. . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
43	42	ifbid 4505	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝑛 → if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)) = if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g‘𝑅)))
44	43	adantl 481	. . . . . . . . . . . . . 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 12422	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
46	45	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
47		fvex 6857	. . . . . . . . . . . . . . . 16 ⊢ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ V
48		fvex 6857	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) ∈ V
49	47, 48	ifex 4532	. . . . . . . . . . . . . . 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 6959	. . . . . . . . . . . . 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 12193	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
53	52	neneqd 2938	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
54	53	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
55	54	iffalsed 4492	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = (0g‘𝑅))
57		eqcom 2744	. . . . . . . . . . . . 13 ⊢ (((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅) ↔ (0g‘𝑅) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
58	57	biimpi 216	. . . . . . . . . . . 12 ⊢ (((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅) → (0g‘𝑅) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
59	56, 58	sylan9eq 2792	. . . . . . . . . . 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 412	. . . . . . . . . 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 3150	. . . . . . . . 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 406	. . . . . . . 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 480	. . . . . . . . 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 22247	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0))
65	64	eqcomd 2743	. . . . . . . . 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 511	. . . . . . 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 412	. . . . . 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 242	. . . . 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 11140	. . . 4 ⊢ 0 ∈ V
73		fveq2 6844	. . . . . 6 ⊢ (𝑛 = 0 → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0))
74		fveq2 6844	. . . . . 6 ⊢ (𝑛 = 0 → ((coe1‘(𝑥𝑀𝑦))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘0))
75	73, 74	eqeq12d 2753	. . . . 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 4852	. . . 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 257	. 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 12416	. . 3 ⊢ ℕ0 = (ℕ ∪ {0})
80	79	raleqi 3296	. 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 234	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ∪ cun 3901  ifcif 4481  {csn 4582   ↦ cmpt 5181  ‘cfv 6502  (class class class)co 7370  Fincfn 8897  0cc0 11040  ℕcn 12159  ℕ₀cn0 12415  Basecbs 17150  0_gc0g 17373  Ringcrg 20185  algSccascl 21824  Poly₁cpl1 22134  coe₁cco1 22135   Mat cmat 22368   ConstPolyMat ccpmat 22664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-of 7634  df-ofr 7635  df-om 7821  df-1st 7945  df-2nd 7946  df-supp 8115  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-er 8647  df-map 8779  df-pm 8780  df-ixp 8850  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-fsupp 9279  df-sup 9359  df-oi 9429  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-z 12503  df-dec 12622  df-uz 12766  df-fz 13438  df-fzo 13585  df-seq 13939  df-hash 14268  df-struct 17088  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-mulr 17205  df-sca 17207  df-vsca 17208  df-ip 17209  df-tset 17210  df-ple 17211  df-ds 17213  df-hom 17215  df-cco 17216  df-0g 17375  df-gsum 17376  df-prds 17381  df-pws 17383  df-mre 17519  df-mrc 17520  df-acs 17522  df-mgm 18579  df-sgrp 18658  df-mnd 18674  df-mhm 18722  df-submnd 18723  df-grp 18883  df-minusg 18884  df-sbg 18885  df-mulg 19015  df-subg 19070  df-ghm 19159  df-cntz 19263  df-cmn 19728  df-abl 19729  df-mgp 20093  df-rng 20105  df-ur 20134  df-ring 20187  df-subrng 20496  df-subrg 20520  df-lmod 20830  df-lss 20900  df-sra 21142  df-rgmod 21143  df-dsmm 21704  df-frlm 21719  df-ascl 21827  df-psr 21882  df-mvr 21883  df-mpl 21884  df-opsr 21886  df-psr1 22137  df-vr1 22138  df-ply1 22139  df-coe1 22140  df-mat 22369  df-cpmat 22667

This theorem is referenced by:  m2cpminvid2  22716