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Theorem m2cpminvid2lem 22643

Description: Lemma for m2cpminvid2 22644. (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 2727	. . . . . . . 8 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
4		eqid 2727	. . . . . . . 8 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
5	1, 2, 3, 4	cpmatelimp 22601	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → (𝑀 ∈ (Base‘(𝑁 Mat 𝑃)) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑘) = (0_g‘𝑅))))
6	5	3impia 1115	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑀 ∈ (Base‘(𝑁 Mat 𝑃)) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑘) = (0_g‘𝑅)))
7	6	simprd 495	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑘) = (0_g‘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) = (0_g‘𝑅))
9		fvoveq1 7437	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (coe₁‘(𝑖𝑀𝑗)) = (coe₁‘(𝑥𝑀𝑗)))
10	9	fveq1d 6893	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → ((coe₁‘(𝑖𝑀𝑗))‘𝑘) = ((coe₁‘(𝑥𝑀𝑗))‘𝑘))
11	10	eqeq1d 2729	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → (((coe₁‘(𝑖𝑀𝑗))‘𝑘) = (0_g‘𝑅) ↔ ((coe₁‘(𝑥𝑀𝑗))‘𝑘) = (0_g‘𝑅)))
12	11	ralbidv 3172	. . . . . . 7 ⊢ (𝑖 = 𝑥 → (∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑘) = (0_g‘𝑅) ↔ ∀𝑘 ∈ ℕ ((coe₁‘(𝑥𝑀𝑗))‘𝑘) = (0_g‘𝑅)))
13		oveq2 7422	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑥𝑀𝑗) = (𝑥𝑀𝑦))
14	13	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (coe₁‘(𝑥𝑀𝑗)) = (coe₁‘(𝑥𝑀𝑦)))
15	14	fveq1d 6893	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((coe₁‘(𝑥𝑀𝑗))‘𝑘) = ((coe₁‘(𝑥𝑀𝑦))‘𝑘))
16	15	eqeq1d 2729	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((coe₁‘(𝑥𝑀𝑗))‘𝑘) = (0_g‘𝑅) ↔ ((coe₁‘(𝑥𝑀𝑦))‘𝑘) = (0_g‘𝑅)))
17	16	ralbidv 3172	. . . . . . 7 ⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ ℕ ((coe₁‘(𝑥𝑀𝑗))‘𝑘) = (0_g‘𝑅) ↔ ∀𝑘 ∈ ℕ ((coe₁‘(𝑥𝑀𝑦))‘𝑘) = (0_g‘𝑅)))
18	12, 17	rspc2v 3618	. . . . . 6 ⊢ ((𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑘) = (0_g‘𝑅) → ∀𝑘 ∈ ℕ ((coe₁‘(𝑥𝑀𝑦))‘𝑘) = (0_g‘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) = (0_g‘𝑅) → ∀𝑘 ∈ ℕ ((coe₁‘(𝑥𝑀𝑦))‘𝑘) = (0_g‘𝑅)))
20		fveqeq2 6900	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (((coe₁‘(𝑥𝑀𝑦))‘𝑘) = (0_g‘𝑅) ↔ ((coe₁‘(𝑥𝑀𝑦))‘𝑛) = (0_g‘𝑅)))
21	20	cbvralvw 3229	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ ((coe₁‘(𝑥𝑀𝑦))‘𝑘) = (0_g‘𝑅) ↔ ∀𝑛 ∈ ℕ ((coe₁‘(𝑥𝑀𝑦))‘𝑛) = (0_g‘𝑅))
22		simpl2 1190	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑅 ∈ Ring)
23		eqid 2727	. . . . . . . . . . . . . . . . . . 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 22599	. . . . . . . . . . . . . . . . . . . 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 22315	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘𝑃))
29		0nn0 12509	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ₀
30		eqid 2727	. . . . . . . . . . . . . . . . . . 19 ⊢ (coe₁‘(𝑥𝑀𝑦)) = (coe₁‘(𝑥𝑀𝑦))
31		eqid 2727	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	30, 23, 2, 31	coe1fvalcl 22118	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥𝑀𝑦) ∈ (Base‘𝑃) ∧ 0 ∈ ℕ₀) → ((coe₁‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
33	28, 29, 32	sylancl 585	. . . . . . . . . . . . . . . . 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 2727	. . . . . . . . . . . . . . . 16 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
37		eqid 2727	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
38	2, 36, 31, 37	coe1scl 22193	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ ((coe₁‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → (coe₁‘((algSc‘𝑃)‘((coe₁‘(𝑥𝑀𝑦))‘0))) = (𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘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), (0_g‘𝑅))))
40	39	fveq1d 6893	. . . . . . . . . . . . 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), (0_g‘𝑅)))‘𝑛))
41		eqidd 2728	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘𝑅))) = (𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘𝑅))))
42		eqeq1 2731	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
43	42	ifbid 4547	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝑛 → if(𝑙 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘𝑅)) = if(𝑛 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘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), (0_g‘𝑅)) = if(𝑛 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘𝑅)))
45		nnnn0 12501	. . . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . . 16 ⊢ ((coe₁‘(𝑥𝑀𝑦))‘0) ∈ V
48		fvex 6904	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝑅) ∈ V
49	47, 48	ifex 4574	. . . . . . . . . . . . . . 15 ⊢ if(𝑛 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘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), (0_g‘𝑅)) ∈ V)
51	41, 44, 46, 50	fvmptd 7006	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ₀ ↦ if(𝑙 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘𝑅)))‘𝑛) = if(𝑛 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘𝑅)))
52		nnne0 12268	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
53	52	neneqd 2940	. . . . . . . . . . . . . . 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 4535	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘𝑅)) = (0_g‘𝑅))
56	40, 51, 55	3eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → ((coe₁‘((algSc‘𝑃)‘((coe₁‘(𝑥𝑀𝑦))‘0)))‘𝑛) = (0_g‘𝑅))
57		eqcom 2734	. . . . . . . . . . . . 13 ⊢ (((coe₁‘(𝑥𝑀𝑦))‘𝑛) = (0_g‘𝑅) ↔ (0_g‘𝑅) = ((coe₁‘(𝑥𝑀𝑦))‘𝑛))
58	57	biimpi 215	. . . . . . . . . . . 12 ⊢ (((coe₁‘(𝑥𝑀𝑦))‘𝑛) = (0_g‘𝑅) → (0_g‘𝑅) = ((coe₁‘(𝑥𝑀𝑦))‘𝑛))
59	56, 58	sylan9eq 2787	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) ∧ ((coe₁‘(𝑥𝑀𝑦))‘𝑛) = (0_g‘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) = (0_g‘𝑅) → ((coe₁‘((algSc‘𝑃)‘((coe₁‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe₁‘(𝑥𝑀𝑦))‘𝑛)))
61	60	ralimdva 3162	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (∀𝑛 ∈ ℕ ((coe₁‘(𝑥𝑀𝑦))‘𝑛) = (0_g‘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) = (0_g‘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) = (0_g‘𝑅)) → (𝑅 ∈ Ring ∧ ((coe₁‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
64	2, 36, 31	ply1sclid 22194	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ ((coe₁‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((coe₁‘(𝑥𝑀𝑦))‘0) = ((coe₁‘((algSc‘𝑃)‘((coe₁‘(𝑥𝑀𝑦))‘0)))‘0))
65	64	eqcomd 2733	. . . . . . . . 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) = (0_g‘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) = (0_g‘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) = (0_g‘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 241	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (∀𝑘 ∈ ℕ ((coe₁‘(𝑥𝑀𝑦))‘𝑘) = (0_g‘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) = (0_g‘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 11230	. . . 4 ⊢ 0 ∈ V
73		fveq2 6891	. . . . . 6 ⊢ (𝑛 = 0 → ((coe₁‘((algSc‘𝑃)‘((coe₁‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe₁‘((algSc‘𝑃)‘((coe₁‘(𝑥𝑀𝑦))‘0)))‘0))
74		fveq2 6891	. . . . . 6 ⊢ (𝑛 = 0 → ((coe₁‘(𝑥𝑀𝑦))‘𝑛) = ((coe₁‘(𝑥𝑀𝑦))‘0))
75	73, 74	eqeq12d 2743	. . . . 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 4890	. . . 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 12495	. . 3 ⊢ ℕ₀ = (ℕ ∪ {0})
80	79	raleqi 3318	. 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 233	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ∀𝑛 ∈ ℕ₀ ((coe₁‘((algSc‘𝑃)‘((coe₁‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe₁‘(𝑥𝑀𝑦))‘𝑛))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 Vcvv 3469 ∪ cun 3942 ifcif 4524 {csn 4624 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 Fincfn 8955 0cc0 11130 ℕcn 12234 ℕ₀cn0 12494 Basecbs 17171 0_gc0g 17412 Ringcrg 20164 algSccascl 21773 Poly₁cpl1 22083 coe₁cco1 22084 Mat cmat 22294 ConstPolyMat ccpmat 22592

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-fzo 13652 df-seq 13991 df-hash 14314 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-subrng 20472 df-subrg 20497 df-lmod 20734 df-lss 20805 df-sra 21047 df-rgmod 21048 df-dsmm 21653 df-frlm 21668 df-ascl 21776 df-psr 21829 df-mvr 21830 df-mpl 21831 df-opsr 21833 df-psr1 22086 df-vr1 22087 df-ply1 22088 df-coe1 22089 df-mat 22295 df-cpmat 22595

This theorem is referenced by: m2cpminvid2 22644

W3C validator