Theorem m2cpminvid2lem 22617

Description: Lemma for m2cpminvid2 22618. (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 2729	. . . . . . . 8 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
4		eqid 2729	. . . . . . . 8 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
5	1, 2, 3, 4	cpmatelimp 22575	. . . . . . 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 1117	. . . . . 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 7392	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑗)))
10	9	fveq1d 6842	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑘) = ((coe1‘(𝑥𝑀𝑗))‘𝑘))
11	10	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → (((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅)))
12	11	ralbidv 3156	. . . . . . 7 ⊢ (𝑖 = 𝑥 → (∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅)))
13		oveq2 7377	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑥𝑀𝑗) = (𝑥𝑀𝑦))
14	13	fveq2d 6844	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (coe1‘(𝑥𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
15	14	fveq1d 6842	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((coe1‘(𝑥𝑀𝑗))‘𝑘) = ((coe1‘(𝑥𝑀𝑦))‘𝑘))
16	15	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅)))
17	16	ralbidv 3156	. . . . . . 7 ⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g‘𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅)))
18	12, 17	rspc2v 3596	. . . . . 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 6849	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅) ↔ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅)))
21	20	cbvralvw 3213	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g‘𝑅) ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g‘𝑅))
22		simpl2 1193	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑅 ∈ Ring)
23		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘𝑃)
24		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
25		simprr 772	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
26	1, 2, 3, 4	cpmatpmat 22573	. . . . . . . . . . . . . . . . . . . 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 22289	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘𝑃))
29		0nn0 12433	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ0
30		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
31		eqid 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	30, 23, 2, 31	coe1fvalcl 22073	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥𝑀𝑦) ∈ (Base‘𝑃) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
33	28, 29, 32	sylancl 586	. . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
37		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) = (0g‘𝑅)
38	2, 36, 31, 37	coe1scl 22149	. . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
43	42	ifbid 4508	. . . . . . . . . . . . . . 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 12425	. . . . . . . . . . . . . . 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 6853	. . . . . . . . . . . . . . . 16 ⊢ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ V
48		fvex 6853	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) ∈ V
49	47, 48	ifex 4535	. . . . . . . . . . . . . . 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 6957	. . . . . . . . . . . . 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 12196	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
53	52	neneqd 2930	. . . . . . . . . . . . . . 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 4495	. . . . . . . . . . . . 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 2768	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = (0g‘𝑅))
57		eqcom 2736	. . . . . . . . . . . . 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 2784	. . . . . . . . . . 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 3145	. . . . . . . . 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 22150	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0))
65	64	eqcomd 2735	. . . . . . . . 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 11144	. . . 4 ⊢ 0 ∈ V
73		fveq2 6840	. . . . . 6 ⊢ (𝑛 = 0 → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0))
74		fveq2 6840	. . . . . 6 ⊢ (𝑛 = 0 → ((coe1‘(𝑥𝑀𝑦))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘0))
75	73, 74	eqeq12d 2745	. . . . 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 4854	. . . 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 12419	. . 3 ⊢ ℕ0 = (ℕ ∪ {0})
80	79	raleqi 3294	. 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 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ∪ cun 3909  ifcif 4484  {csn 4585   ↦ cmpt 5183  ‘cfv 6499  (class class class)co 7369  Fincfn 8895  0cc0 11044  ℕcn 12162  ℕ₀cn0 12418  Basecbs 17155  0_gc0g 17378  Ringcrg 20118  algSccascl 21737  Poly₁cpl1 22037  coe₁cco1 22038   Mat cmat 22270   ConstPolyMat ccpmat 22566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-ofr 7634  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-sup 9369  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-fzo 13592  df-seq 13943  df-hash 14272  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-hom 17220  df-cco 17221  df-0g 17380  df-gsum 17381  df-prds 17386  df-pws 17388  df-mre 17523  df-mrc 17524  df-acs 17526  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-mhm 18686  df-submnd 18687  df-grp 18844  df-minusg 18845  df-sbg 18846  df-mulg 18976  df-subg 19031  df-ghm 19121  df-cntz 19225  df-cmn 19688  df-abl 19689  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-subrng 20431  df-subrg 20455  df-lmod 20744  df-lss 20814  df-sra 21056  df-rgmod 21057  df-dsmm 21617  df-frlm 21632  df-ascl 21740  df-psr 21794  df-mvr 21795  df-mpl 21796  df-opsr 21798  df-psr1 22040  df-vr1 22041  df-ply1 22042  df-coe1 22043  df-mat 22271  df-cpmat 22569

This theorem is referenced by:  m2cpminvid2  22618