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Theorem m2cpminvid2 22032
Description: The transformation applied to the inverse transformation of a constant polynomial matrix over the ring 𝑅 results in the matrix itself. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)
Hypotheses
Ref Expression
m2cpminvid2.s 𝑆 = (𝑁 ConstPolyMat 𝑅)
m2cpminvid2.i 𝐼 = (𝑁 cPolyMatToMat 𝑅)
m2cpminvid2.t 𝑇 = (𝑁 matToPolyMat 𝑅)
Assertion
Ref Expression
m2cpminvid2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑇‘(𝐼𝑀)) = 𝑀)

Proof of Theorem m2cpminvid2
Dummy variables 𝑖 𝑗 𝑥 𝑦 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 m2cpminvid2.i . . . 4 𝐼 = (𝑁 cPolyMatToMat 𝑅)
2 m2cpminvid2.s . . . 4 𝑆 = (𝑁 ConstPolyMat 𝑅)
31, 2cpm2mval 22027 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝐼𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
43fveq2d 6842 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑇‘(𝐼𝑀)) = (𝑇‘(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))))
5 simp1 1137 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → 𝑁 ∈ Fin)
6 simp2 1138 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → 𝑅 ∈ Ring)
7 eqid 2738 . . . . 5 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
8 eqid 2738 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
9 eqid 2738 . . . . 5 (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
10 eqid 2738 . . . . . . 7 (𝑁 Mat (Poly1𝑅)) = (𝑁 Mat (Poly1𝑅))
11 eqid 2738 . . . . . . 7 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
12 eqid 2738 . . . . . . 7 (Base‘(𝑁 Mat (Poly1𝑅))) = (Base‘(𝑁 Mat (Poly1𝑅)))
13 simp2 1138 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁𝑦𝑁) → 𝑥𝑁)
14 simp3 1139 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁𝑦𝑁) → 𝑦𝑁)
15 eqid 2738 . . . . . . . . 9 (Poly1𝑅) = (Poly1𝑅)
162, 15, 10, 12cpmatpmat 21987 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1𝑅))))
17163ad2ant1 1134 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁𝑦𝑁) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1𝑅))))
1810, 11, 12, 13, 14, 17matecld 21703 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1𝑅)))
19 0nn0 12362 . . . . . 6 0 ∈ ℕ0
20 eqid 2738 . . . . . . 7 (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
2120, 11, 15, 8coe1fvalcl 21511 . . . . . 6 (((𝑥𝑀𝑦) ∈ (Base‘(Poly1𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
2218, 19, 21sylancl 587 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁𝑦𝑁) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
237, 8, 9, 5, 6, 22matbas2d 21700 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(𝑁 Mat 𝑅)))
24 m2cpminvid2.t . . . . 5 𝑇 = (𝑁 matToPolyMat 𝑅)
25 eqid 2738 . . . . 5 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
2624, 7, 9, 15, 25mat2pmatval 22001 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘(𝑖(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗))))
275, 6, 23, 26syl3anc 1372 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑇‘(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘(𝑖(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗))))
28 eqidd 2739 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
29 oveq12 7359 . . . . . . . . 9 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
3029fveq2d 6842 . . . . . . . 8 ((𝑥 = 𝑖𝑦 = 𝑗) → (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑖𝑀𝑗)))
3130fveq1d 6840 . . . . . . 7 ((𝑥 = 𝑖𝑦 = 𝑗) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑖𝑀𝑗))‘0))
3231adantl 483 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) ∧ (𝑥 = 𝑖𝑦 = 𝑗)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑖𝑀𝑗))‘0))
33 simp2 1138 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
34 simp3 1139 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
35 fvexd 6853 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ V)
3628, 32, 33, 34, 35ovmpod 7500 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗) = ((coe1‘(𝑖𝑀𝑗))‘0))
3736fveq2d 6842 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘(Poly1𝑅))‘(𝑖(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗)) = ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))
3837mpoeq3dva 7427 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘(𝑖(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))))
3927, 38eqtrd 2778 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑇‘(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))))
402, 15m2cpminvid2lem 22031 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
41 simpl2 1193 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑅 ∈ Ring)
42 simprl 770 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
43 simprr 772 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑦𝑁)
4416adantr 482 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1𝑅))))
4510, 11, 12, 42, 43, 44matecld 21703 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1𝑅)))
4645, 19, 21sylancl 587 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
4715, 25, 8, 11ply1sclcl 21585 . . . . . . . 8 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1𝑅)))
4841, 46, 47syl2anc 585 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1𝑅)))
49 eqid 2738 . . . . . . . . 9 (coe1‘((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0))) = (coe1‘((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
5015, 11, 49, 20ply1coe1eq 21597 . . . . . . . 8 ((𝑅 ∈ Ring ∧ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1𝑅)) ∧ (𝑥𝑀𝑦) ∈ (Base‘(Poly1𝑅))) → (∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛) ↔ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦)))
5150bicomd 222 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1𝑅)) ∧ (𝑥𝑀𝑦) ∈ (Base‘(Poly1𝑅))) → (((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛)))
5241, 48, 45, 51syl3anc 1372 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛)))
5340, 52mpbird 257 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦))
5453ralrimivva 3196 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → ∀𝑥𝑁𝑦𝑁 ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦))
55 eqidd 2739 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))))
56 oveq12 7359 . . . . . . . . . . . 12 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
5756fveq2d 6842 . . . . . . . . . . 11 ((𝑖 = 𝑥𝑗 = 𝑦) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
5857fveq1d 6840 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = 𝑦) → ((coe1‘(𝑖𝑀𝑗))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
5958fveq2d 6842 . . . . . . . . 9 ((𝑖 = 𝑥𝑗 = 𝑦) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) = ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
6059adantl 483 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) ∧ (𝑖 = 𝑥𝑗 = 𝑦)) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) = ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
61 simplr 768 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → 𝑥𝑁)
62 simpr 486 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → 𝑦𝑁)
63 fvexd 6853 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ V)
6455, 60, 61, 62, 63ovmpod 7500 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
6564eqeq1d 2740 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → ((𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦) ↔ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦)))
6665anasss 468 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ((𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦) ↔ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦)))
67662ralbidva 3209 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦) ↔ ∀𝑥𝑁𝑦𝑁 ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦)))
6854, 67mpbird 257 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦))
69 fvexd 6853 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (Poly1𝑅) ∈ V)
7063ad2ant1 1134 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
71163ad2ant1 1134 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1𝑅))))
7210, 11, 12, 33, 34, 71matecld 21703 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Poly1𝑅)))
73 eqid 2738 . . . . . . . 8 (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
7473, 11, 15, 8coe1fvalcl 21511 . . . . . . 7 (((𝑖𝑀𝑗) ∈ (Base‘(Poly1𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅))
7572, 19, 74sylancl 587 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅))
7615, 25, 8, 11ply1sclcl 21585 . . . . . 6 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅)) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) ∈ (Base‘(Poly1𝑅)))
7770, 75, 76syl2anc 585 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) ∈ (Base‘(Poly1𝑅)))
7810, 11, 12, 5, 69, 77matbas2d 21700 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) ∈ (Base‘(𝑁 Mat (Poly1𝑅))))
7910, 12eqmat 21701 . . . 4 (((𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) ∈ (Base‘(𝑁 Mat (Poly1𝑅))) ∧ 𝑀 ∈ (Base‘(𝑁 Mat (Poly1𝑅)))) → ((𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = 𝑀 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦)))
8078, 16, 79syl2anc 585 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → ((𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = 𝑀 ↔ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦)))
8168, 80mpbird 257 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = 𝑀)
824, 39, 813eqtrd 2782 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑇‘(𝐼𝑀)) = 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3063  Vcvv 3444  cfv 6492  (class class class)co 7350  cmpo 7352  Fincfn 8817  0cc0 10985  0cn0 12347  Basecbs 17019  Ringcrg 19894  algSccascl 21187  Poly1cpl1 21476  coe1cco1 21477   Mat cmat 21682   ConstPolyMat ccpmat 21980   matToPolyMat cmat2pmat 21981   cPolyMatToMat ccpmat2mat 21982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663  ax-cnex 11041  ax-resscn 11042  ax-1cn 11043  ax-icn 11044  ax-addcl 11045  ax-addrcl 11046  ax-mulcl 11047  ax-mulrcl 11048  ax-mulcom 11049  ax-addass 11050  ax-mulass 11051  ax-distr 11052  ax-i2m1 11053  ax-1ne0 11054  ax-1rid 11055  ax-rnegex 11056  ax-rrecex 11057  ax-cnre 11058  ax-pre-lttri 11059  ax-pre-lttrn 11060  ax-pre-ltadd 11061  ax-pre-mulgt0 11062
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4865  df-int 4907  df-iun 4955  df-iin 4956  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  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 6250  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7306  df-ov 7353  df-oprab 7354  df-mpo 7355  df-of 7608  df-ofr 7609  df-om 7794  df-1st 7912  df-2nd 7913  df-supp 8061  df-frecs 8180  df-wrecs 8211  df-recs 8285  df-rdg 8324  df-1o 8380  df-er 8582  df-map 8701  df-pm 8702  df-ixp 8770  df-en 8818  df-dom 8819  df-sdom 8820  df-fin 8821  df-fsupp 9240  df-sup 9312  df-oi 9380  df-card 9809  df-pnf 11125  df-mnf 11126  df-xr 11127  df-ltxr 11128  df-le 11129  df-sub 11321  df-neg 11322  df-nn 12088  df-2 12150  df-3 12151  df-4 12152  df-5 12153  df-6 12154  df-7 12155  df-8 12156  df-9 12157  df-n0 12348  df-z 12434  df-dec 12553  df-uz 12698  df-fz 13355  df-fzo 13498  df-seq 13837  df-hash 14160  df-struct 16955  df-sets 16972  df-slot 16990  df-ndx 17002  df-base 17020  df-ress 17049  df-plusg 17082  df-mulr 17083  df-sca 17085  df-vsca 17086  df-ip 17087  df-tset 17088  df-ple 17089  df-ds 17091  df-hom 17093  df-cco 17094  df-0g 17259  df-gsum 17260  df-prds 17265  df-pws 17267  df-mre 17402  df-mrc 17403  df-acs 17405  df-mgm 18433  df-sgrp 18482  df-mnd 18493  df-mhm 18537  df-submnd 18538  df-grp 18687  df-minusg 18688  df-sbg 18689  df-mulg 18808  df-subg 18860  df-ghm 18941  df-cntz 19032  df-cmn 19499  df-abl 19500  df-mgp 19832  df-ur 19849  df-srg 19853  df-ring 19896  df-subrg 20149  df-lmod 20253  df-lss 20322  df-sra 20562  df-rgmod 20563  df-dsmm 21067  df-frlm 21082  df-ascl 21190  df-psr 21240  df-mvr 21241  df-mpl 21242  df-opsr 21244  df-psr1 21479  df-vr1 21480  df-ply1 21481  df-coe1 21482  df-mat 21683  df-cpmat 21983  df-mat2pmat 21984  df-cpmat2mat 21985
This theorem is referenced by:  m2cpmfo  22033  m2cpminv  22037  cpmadumatpoly  22160  cayhamlem4  22165
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