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Theorem m2cpminvid2lem 22103
Description: Lemma for m2cpminvid2 22104. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)
Hypotheses
Ref Expression
m2cpminvid2lem.s 𝑆 = (𝑁 ConstPolyMat 𝑅)
m2cpminvid2lem.p 𝑃 = (Poly1𝑅)
Assertion
Ref Expression
m2cpminvid2lem (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ∀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.
StepHypRef Expression
1 m2cpminvid2lem.s . . . . . . . 8 𝑆 = (𝑁 ConstPolyMat 𝑅)
2 m2cpminvid2lem.p . . . . . . . 8 𝑃 = (Poly1𝑅)
3 eqid 2736 . . . . . . . 8 (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
4 eqid 2736 . . . . . . . 8 (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
51, 2, 3, 4cpmatelimp 22061 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀𝑆 → (𝑀 ∈ (Base‘(𝑁 Mat 𝑃)) ∧ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅))))
653impia 1117 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑀 ∈ (Base‘(𝑁 Mat 𝑃)) ∧ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
76simprd 496 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅))
87adantr 481 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅))
9 fvoveq1 7380 . . . . . . . . . 10 (𝑖 = 𝑥 → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑗)))
109fveq1d 6844 . . . . . . . . 9 (𝑖 = 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑘) = ((coe1‘(𝑥𝑀𝑗))‘𝑘))
1110eqeq1d 2738 . . . . . . . 8 (𝑖 = 𝑥 → (((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅) ↔ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g𝑅)))
1211ralbidv 3174 . . . . . . 7 (𝑖 = 𝑥 → (∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g𝑅)))
13 oveq2 7365 . . . . . . . . . . 11 (𝑗 = 𝑦 → (𝑥𝑀𝑗) = (𝑥𝑀𝑦))
1413fveq2d 6846 . . . . . . . . . 10 (𝑗 = 𝑦 → (coe1‘(𝑥𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
1514fveq1d 6844 . . . . . . . . 9 (𝑗 = 𝑦 → ((coe1‘(𝑥𝑀𝑗))‘𝑘) = ((coe1‘(𝑥𝑀𝑦))‘𝑘))
1615eqeq1d 2738 . . . . . . . 8 (𝑗 = 𝑦 → (((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g𝑅) ↔ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅)))
1716ralbidv 3174 . . . . . . 7 (𝑗 = 𝑦 → (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅)))
1812, 17rspc2v 3590 . . . . . 6 ((𝑥𝑁𝑦𝑁) → (∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅) → ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅)))
1918adantl 482 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅) → ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅)))
20 fveqeq2 6851 . . . . . . 7 (𝑘 = 𝑛 → (((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅) ↔ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅)))
2120cbvralvw 3225 . . . . . 6 (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅) ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅))
22 simpl2 1192 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑅 ∈ Ring)
23 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑃) = (Base‘𝑃)
24 simprl 769 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
25 simprr 771 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑦𝑁)
261, 2, 3, 4cpmatpmat 22059 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → 𝑀 ∈ (Base‘(𝑁 Mat 𝑃)))
2726adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑀 ∈ (Base‘(𝑁 Mat 𝑃)))
283, 23, 4, 24, 25, 27matecld 21775 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘𝑃))
29 0nn0 12428 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
30 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
31 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) = (Base‘𝑅)
3230, 23, 2, 31coe1fvalcl 21583 . . . . . . . . . . . . . . . . . 18 (((𝑥𝑀𝑦) ∈ (Base‘𝑃) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
3328, 29, 32sylancl 586 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
3422, 33jca 512 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
3534adantr 481 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → (𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
36 eqid 2736 . . . . . . . . . . . . . . . 16 (algSc‘𝑃) = (algSc‘𝑃)
37 eqid 2736 . . . . . . . . . . . . . . . 16 (0g𝑅) = (0g𝑅)
382, 36, 31, 37coe1scl 21658 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → (coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅))))
3935, 38syl 17 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → (coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅))))
4039fveq1d 6844 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)))‘𝑛))
41 eqidd 2737 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅))))
42 eqeq1 2740 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
4342ifbid 4509 . . . . . . . . . . . . . . 15 (𝑙 = 𝑛 → if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)) = if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)))
4443adantl 482 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 = 𝑛) → if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)) = if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)))
45 nnnn0 12420 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
4645adantl 482 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
47 fvex 6855 . . . . . . . . . . . . . . . 16 ((coe1‘(𝑥𝑀𝑦))‘0) ∈ V
48 fvex 6855 . . . . . . . . . . . . . . . 16 (0g𝑅) ∈ V
4947, 48ifex 4536 . . . . . . . . . . . . . . 15 if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)) ∈ V
5049a1i 11 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)) ∈ V)
5141, 44, 46, 50fvmptd 6955 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)))‘𝑛) = if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)))
52 nnne0 12187 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
5352neneqd 2948 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
5453adantl 482 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
5554iffalsed 4497 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)) = (0g𝑅))
5640, 51, 553eqtrd 2780 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = (0g𝑅))
57 eqcom 2743 . . . . . . . . . . . . 13 (((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅) ↔ (0g𝑅) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
5857biimpi 215 . . . . . . . . . . . 12 (((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅) → (0g𝑅) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
5956, 58sylan9eq 2796 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅)) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
6059ex 413 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → (((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛)))
6160ralimdva 3164 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅) → ∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛)))
6261imp 407 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅)) → ∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
6334adantr 481 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅)) → (𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
642, 36, 31ply1sclid 21659 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0))
6564eqcomd 2742 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
6663, 65syl 17 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅)) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
6762, 66jca 512 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅)) → (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0)))
6867ex 413 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅) → (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))))
6921, 68biimtrid 241 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅) → (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))))
7019, 69syld 47 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅) → (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))))
718, 70mpd 15 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (∀𝑛 ∈ ℕ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ∧ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0)))
72 c0ex 11149 . . . 4 0 ∈ V
73 fveq2 6842 . . . . . 6 (𝑛 = 0 → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0))
74 fveq2 6842 . . . . . 6 (𝑛 = 0 → ((coe1‘(𝑥𝑀𝑦))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘0))
7573, 74eqeq12d 2752 . . . . 5 (𝑛 = 0 → (((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ↔ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0)))
7675ralunsn 4851 . . . 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))))
7772, 76mp1i 13 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (∀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))))
7871, 77mpbird 256 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ∀𝑛 ∈ (ℕ ∪ {0})((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
79 df-n0 12414 . . 3 0 = (ℕ ∪ {0})
8079raleqi 3311 . 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))
8178, 80sylibr 233 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ∀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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cun 3908  ifcif 4486  {csn 4586  cmpt 5188  cfv 6496  (class class class)co 7357  Fincfn 8883  0cc0 11051  cn 12153  0cn0 12413  Basecbs 17083  0gc0g 17321  Ringcrg 19964  algSccascl 21258  Poly1cpl1 21548  coe1cco1 21549   Mat cmat 21754   ConstPolyMat ccpmat 22052
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-subrg 20220  df-lmod 20324  df-lss 20393  df-sra 20633  df-rgmod 20634  df-dsmm 21138  df-frlm 21153  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-vr1 21552  df-ply1 21553  df-coe1 21554  df-mat 21755  df-cpmat 22055
This theorem is referenced by:  m2cpminvid2  22104
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