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Theorem m2cpminvid2lem 20607
 Description: Lemma for m2cpminvid2 20608. (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 2651 . . . . . . . 8 (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃)
4 eqid 2651 . . . . . . . 8 (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃))
51, 2, 3, 4cpmatelimp 20565 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀𝑆 → (𝑀 ∈ (Base‘(𝑁 Mat 𝑃)) ∧ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅))))
653impia 1280 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑀 ∈ (Base‘(𝑁 Mat 𝑃)) ∧ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
7 simpr 476 . . . . . 6 ((𝑀 ∈ (Base‘(𝑁 Mat 𝑃)) ∧ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)) → ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅))
86, 7syl 17 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅))
98adantr 480 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅))
10 oveq1 6697 . . . . . . . . . . 11 (𝑖 = 𝑥 → (𝑖𝑀𝑗) = (𝑥𝑀𝑗))
1110fveq2d 6233 . . . . . . . . . 10 (𝑖 = 𝑥 → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑗)))
1211fveq1d 6231 . . . . . . . . 9 (𝑖 = 𝑥 → ((coe1‘(𝑖𝑀𝑗))‘𝑘) = ((coe1‘(𝑥𝑀𝑗))‘𝑘))
1312eqeq1d 2653 . . . . . . . 8 (𝑖 = 𝑥 → (((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅) ↔ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g𝑅)))
1413ralbidv 3015 . . . . . . 7 (𝑖 = 𝑥 → (∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g𝑅)))
15 oveq2 6698 . . . . . . . . . . 11 (𝑗 = 𝑦 → (𝑥𝑀𝑗) = (𝑥𝑀𝑦))
1615fveq2d 6233 . . . . . . . . . 10 (𝑗 = 𝑦 → (coe1‘(𝑥𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
1716fveq1d 6231 . . . . . . . . 9 (𝑗 = 𝑦 → ((coe1‘(𝑥𝑀𝑗))‘𝑘) = ((coe1‘(𝑥𝑀𝑦))‘𝑘))
1817eqeq1d 2653 . . . . . . . 8 (𝑗 = 𝑦 → (((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g𝑅) ↔ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅)))
1918ralbidv 3015 . . . . . . 7 (𝑗 = 𝑦 → (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑗))‘𝑘) = (0g𝑅) ↔ ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅)))
2014, 19rspc2v 3353 . . . . . 6 ((𝑥𝑁𝑦𝑁) → (∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅) → ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅)))
2120adantl 481 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅) → ∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅)))
22 fveq2 6229 . . . . . . . 8 (𝑘 = 𝑛 → ((coe1‘(𝑥𝑀𝑦))‘𝑘) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
2322eqeq1d 2653 . . . . . . 7 (𝑘 = 𝑛 → (((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅) ↔ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅)))
2423cbvralv 3201 . . . . . 6 (∀𝑘 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑘) = (0g𝑅) ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅))
25 simpl2 1085 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑅 ∈ Ring)
26 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑃) = (Base‘𝑃)
27 simprl 809 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
28 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((𝑥𝑁𝑦𝑁) → 𝑦𝑁)
2928adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑦𝑁)
301, 2, 3, 4cpmatpmat 20563 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → 𝑀 ∈ (Base‘(𝑁 Mat 𝑃)))
3130adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑀 ∈ (Base‘(𝑁 Mat 𝑃)))
323, 26, 4, 27, 29, 31matecld 20280 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘𝑃))
33 0nn0 11345 . . . . . . . . . . . . . . . . . 18 0 ∈ ℕ0
34 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
35 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) = (Base‘𝑅)
3634, 26, 2, 35coe1fvalcl 19630 . . . . . . . . . . . . . . . . . 18 (((𝑥𝑀𝑦) ∈ (Base‘𝑃) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
3732, 33, 36sylancl 695 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
3825, 37jca 553 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
3938adantr 480 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → (𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
40 eqid 2651 . . . . . . . . . . . . . . . 16 (algSc‘𝑃) = (algSc‘𝑃)
41 eqid 2651 . . . . . . . . . . . . . . . 16 (0g𝑅) = (0g𝑅)
422, 40, 35, 41coe1scl 19705 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → (coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅))))
4339, 42syl 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))))
4443fveq1d 6231 . . . . . . . . . . . . 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))
45 eqidd 2652 . . . . . . . . . . . . . 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))))
46 eqeq1 2655 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
4746ifbid 4141 . . . . . . . . . . . . . . 15 (𝑙 = 𝑛 → if(𝑙 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)) = if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)))
4847adantl 481 . . . . . . . . . . . . . 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)))
49 nnnn0 11337 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
5049adantl 481 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
51 fvex 6239 . . . . . . . . . . . . . . . 16 ((coe1‘(𝑥𝑀𝑦))‘0) ∈ V
52 fvex 6239 . . . . . . . . . . . . . . . 16 (0g𝑅) ∈ V
5351, 52ifex 4189 . . . . . . . . . . . . . . 15 if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)) ∈ V
5453a1i 11 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)) ∈ V)
5545, 48, 50, 54fvmptd 6327 . . . . . . . . . . . . 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)))
56 nnne0 11091 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑛 ≠ 0)
5756neneqd 2828 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
5857adantl 481 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
5958iffalsed 4130 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → if(𝑛 = 0, ((coe1‘(𝑥𝑀𝑦))‘0), (0g𝑅)) = (0g𝑅))
6044, 55, 593eqtrd 2689 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = (0g𝑅))
61 eqcom 2658 . . . . . . . . . . . . 13 (((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅) ↔ (0g𝑅) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
6261biimpi 206 . . . . . . . . . . . 12 (((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅) → (0g𝑅) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
6360, 62sylan9eq 2705 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅)) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
6463ex 449 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑛 ∈ ℕ) → (((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛)))
6564ralimdva 2991 . . . . . . . . 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)))
6665imp 444 . . . . . . . 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))
6738adantr 480 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅)) → (𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)))
682, 40, 35ply1sclid 19706 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0))
6968eqcomd 2657 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
7067, 69syl 17 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) ∧ ∀𝑛 ∈ ℕ ((coe1‘(𝑥𝑀𝑦))‘𝑛) = (0g𝑅)) → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
7166, 70jca 553 . . . . . . 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)))
7271ex 449 . . . . . 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))))
7324, 72syl5bi 232 . . . . 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))))
7421, 73syld 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))))
759, 74mpd 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)))
76 c0ex 10072 . . . 4 0 ∈ V
77 fveq2 6229 . . . . . 6 (𝑛 = 0 → ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0))
78 fveq2 6229 . . . . . 6 (𝑛 = 0 → ((coe1‘(𝑥𝑀𝑦))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘0))
7977, 78eqeq12d 2666 . . . . 5 (𝑛 = 0 → (((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ↔ ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0)))
8079ralunsn 4454 . . . 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))))
8176, 80mp1i 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))))
8275, 81mpbird 247 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ∀𝑛 ∈ (ℕ ∪ {0})((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
83 df-n0 11331 . . 3 0 = (ℕ ∪ {0})
8483raleqi 3172 . 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))
8582, 84sylibr 224 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 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ∪ cun 3605  ifcif 4119  {csn 4210   ↦ cmpt 4762  ‘cfv 5926  (class class class)co 6690  Fincfn 7997  0cc0 9974  ℕcn 11058  ℕ0cn0 11330  Basecbs 15904  0gc0g 16147  Ringcrg 18593  algSccascl 19359  Poly1cpl1 19595  coe1cco1 19596   Mat cmat 20261   ConstPolyMat ccpmat 20556 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-0g 16149  df-gsum 16150  df-prds 16155  df-pws 16157  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-subrg 18826  df-lmod 18913  df-lss 18981  df-sra 19220  df-rgmod 19221  df-ascl 19362  df-psr 19404  df-mvr 19405  df-mpl 19406  df-opsr 19408  df-psr1 19598  df-vr1 19599  df-ply1 19600  df-coe1 19601  df-dsmm 20124  df-frlm 20139  df-mat 20262  df-cpmat 20559 This theorem is referenced by:  m2cpminvid2  20608
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