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Theorem m2cpminvid2lem 22686

Description: Lemma for m2cpminvid2 22687. (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 2725	. . . . . . . 8 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
4		eqid 2725	. . . . . . . 8 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
5	1, 2, 3, 4	cpmatelimp 22644	. . . . . . 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 1114	. . . . . 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 494	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑘) = (0_g‘𝑅))
8	7	adantr 479	. . . 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 7440	. . . . . . . . . 10 ⊢ (𝑖 = 𝑥 → (coe₁‘(𝑖𝑀𝑗)) = (coe₁‘(𝑥𝑀𝑗)))
10	9	fveq1d 6896	. . . . . . . . 9 ⊢ (𝑖 = 𝑥 → ((coe₁‘(𝑖𝑀𝑗))‘𝑘) = ((coe₁‘(𝑥𝑀𝑗))‘𝑘))
11	10	eqeq1d 2727	. . . . . . . 8 ⊢ (𝑖 = 𝑥 → (((coe₁‘(𝑖𝑀𝑗))‘𝑘) = (0_g‘𝑅) ↔ ((coe₁‘(𝑥𝑀𝑗))‘𝑘) = (0_g‘𝑅)))
12	11	ralbidv 3168	. . . . . . 7 ⊢ (𝑖 = 𝑥 → (∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑘) = (0_g‘𝑅) ↔ ∀𝑘 ∈ ℕ ((coe₁‘(𝑥𝑀𝑗))‘𝑘) = (0_g‘𝑅)))
13		oveq2 7425	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑥𝑀𝑗) = (𝑥𝑀𝑦))
14	13	fveq2d 6898	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (coe₁‘(𝑥𝑀𝑗)) = (coe₁‘(𝑥𝑀𝑦)))
15	14	fveq1d 6896	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → ((coe₁‘(𝑥𝑀𝑗))‘𝑘) = ((coe₁‘(𝑥𝑀𝑦))‘𝑘))
16	15	eqeq1d 2727	. . . . . . . 8 ⊢ (𝑗 = 𝑦 → (((coe₁‘(𝑥𝑀𝑗))‘𝑘) = (0_g‘𝑅) ↔ ((coe₁‘(𝑥𝑀𝑦))‘𝑘) = (0_g‘𝑅)))
17	16	ralbidv 3168	. . . . . . 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 480	. . . . 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 6903	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (((coe₁‘(𝑥𝑀𝑦))‘𝑘) = (0_g‘𝑅) ↔ ((coe₁‘(𝑥𝑀𝑦))‘𝑛) = (0_g‘𝑅)))
21	20	cbvralvw 3225	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ ((coe₁‘(𝑥𝑀𝑦))‘𝑘) = (0_g‘𝑅) ↔ ∀𝑛 ∈ ℕ ((coe₁‘(𝑥𝑀𝑦))‘𝑛) = (0_g‘𝑅))
22		simpl2 1189	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑅 ∈ Ring)
23		eqid 2725	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘𝑃)
24		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
25		simprr 771	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
26	1, 2, 3, 4	cpmatpmat 22642	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ (Base‘(𝑁 Mat 𝑃)))
27	26	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑀 ∈ (Base‘(𝑁 Mat 𝑃)))
28	3, 23, 4, 24, 25, 27	matecld 22358	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘𝑃))
29		0nn0 12517	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ₀
30		eqid 2725	. . . . . . . . . . . . . . . . . . 19 ⊢ (coe₁‘(𝑥𝑀𝑦)) = (coe₁‘(𝑥𝑀𝑦))
31		eqid 2725	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	30, 23, 2, 31	coe1fvalcl 22140	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥𝑀𝑦) ∈ (Base‘𝑃) ∧ 0 ∈ ℕ₀) → ((coe₁‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
33	28, 29, 32	sylancl 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((coe₁‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
34	22, 33	jca 510	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑅 ∈ Ring ∧ ((coe₁‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
35	34	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → (𝑅 ∈ Ring ∧ ((coe₁‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
36		eqid 2725	. . . . . . . . . . . . . . . 16 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
37		eqid 2725	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
38	2, 36, 31, 37	coe1scl 22215	. . . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
43	42	ifbid 4552	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝑛 → if(𝑙 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘𝑅)) = if(𝑛 = 0, ((coe₁‘(𝑥𝑀𝑦))‘0), (0_g‘𝑅)))
44	43	adantl 480	. . . . . . . . . . . . . 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 12509	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
46	45	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
47		fvex 6907	. . . . . . . . . . . . . . . 16 ⊢ ((coe₁‘(𝑥𝑀𝑦))‘0) ∈ V
48		fvex 6907	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝑅) ∈ V
49	47, 48	ifex 4579	. . . . . . . . . . . . . . 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 7009	. . . . . . . . . . . . 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 12276	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
53	52	neneqd 2935	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
54	53	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
55	54	iffalsed 4540	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . 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 2732	. . . . . . . . . . . . 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 2785	. . . . . . . . . . 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 411	. . . . . . . . . 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 3157	. . . . . . . . 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 405	. . . . . . . 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 479	. . . . . . . . 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 22216	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ ((coe₁‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((coe₁‘(𝑥𝑀𝑦))‘0) = ((coe₁‘((algSc‘𝑃)‘((coe₁‘(𝑥𝑀𝑦))‘0)))‘0))
65	64	eqcomd 2731	. . . . . . . . 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 510	. . . . . . 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 411	. . . . . 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 11238	. . . 4 ⊢ 0 ∈ V
73		fveq2 6894	. . . . . 6 ⊢ (𝑛 = 0 → ((coe₁‘((algSc‘𝑃)‘((coe₁‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe₁‘((algSc‘𝑃)‘((coe₁‘(𝑥𝑀𝑦))‘0)))‘0))
74		fveq2 6894	. . . . . 6 ⊢ (𝑛 = 0 → ((coe₁‘(𝑥𝑀𝑦))‘𝑛) = ((coe₁‘(𝑥𝑀𝑦))‘0))
75	73, 74	eqeq12d 2741	. . . . 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 4895	. . . 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 256	. 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 12503	. . 3 ⊢ ℕ₀ = (ℕ ∪ {0})
80	79	raleqi 3313	. 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 Vcvv 3463 ∪ cun 3943 ifcif 4529 {csn 4629 ↦ cmpt 5231 ‘cfv 6547 (class class class)co 7417 Fincfn 8962 0cc0 11138 ℕcn 12242 ℕ₀cn0 12502 Basecbs 17179 0_gc0g 17420 Ringcrg 20177 algSccascl 21790 Poly₁cpl1 22104 coe₁cco1 22105 Mat cmat 22337 ConstPolyMat ccpmat 22635

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 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 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-ofr 7684 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 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-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-subrng 20487 df-subrg 20512 df-lmod 20749 df-lss 20820 df-sra 21062 df-rgmod 21063 df-dsmm 21670 df-frlm 21685 df-ascl 21793 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-psr1 22107 df-vr1 22108 df-ply1 22109 df-coe1 22110 df-mat 22338 df-cpmat 22638

This theorem is referenced by: m2cpminvid2 22687

W3C validator