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Theorem m2cpminvid2 20762
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 20757 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝐼𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
43fveq2d 6356 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑇‘(𝐼𝑀)) = (𝑇‘(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))))
5 simp1 1131 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → 𝑁 ∈ Fin)
6 simp2 1132 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → 𝑅 ∈ Ring)
7 eqid 2760 . . . . 5 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
8 eqid 2760 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
9 eqid 2760 . . . . 5 (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
10 eqid 2760 . . . . . . 7 (𝑁 Mat (Poly1𝑅)) = (𝑁 Mat (Poly1𝑅))
11 eqid 2760 . . . . . . 7 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
12 eqid 2760 . . . . . . 7 (Base‘(𝑁 Mat (Poly1𝑅))) = (Base‘(𝑁 Mat (Poly1𝑅)))
13 simp2 1132 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁𝑦𝑁) → 𝑥𝑁)
14 simp3 1133 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁𝑦𝑁) → 𝑦𝑁)
15 eqid 2760 . . . . . . . . 9 (Poly1𝑅) = (Poly1𝑅)
162, 15, 10, 12cpmatpmat 20717 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1𝑅))))
17163ad2ant1 1128 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁𝑦𝑁) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1𝑅))))
1810, 11, 12, 13, 14, 17matecld 20434 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁𝑦𝑁) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1𝑅)))
19 0nn0 11499 . . . . . 6 0 ∈ ℕ0
20 eqid 2760 . . . . . . 7 (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑥𝑀𝑦))
2120, 11, 15, 8coe1fvalcl 19784 . . . . . 6 (((𝑥𝑀𝑦) ∈ (Base‘(Poly1𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
2218, 19, 21sylancl 697 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁𝑦𝑁) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
237, 8, 9, 5, 6, 22matbas2d 20431 . . . 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 2760 . . . . 5 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
2624, 7, 9, 15, 25mat2pmatval 20731 . . . 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 1477 . . 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 2761 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
29 oveq12 6822 . . . . . . . . 9 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑥𝑀𝑦) = (𝑖𝑀𝑗))
3029fveq2d 6356 . . . . . . . 8 ((𝑥 = 𝑖𝑦 = 𝑗) → (coe1‘(𝑥𝑀𝑦)) = (coe1‘(𝑖𝑀𝑗)))
3130fveq1d 6354 . . . . . . 7 ((𝑥 = 𝑖𝑦 = 𝑗) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑖𝑀𝑗))‘0))
3231adantl 473 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) ∧ (𝑥 = 𝑖𝑦 = 𝑗)) → ((coe1‘(𝑥𝑀𝑦))‘0) = ((coe1‘(𝑖𝑀𝑗))‘0))
33 simp2 1132 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
34 simp3 1133 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
35 fvexd 6364 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ V)
3628, 32, 33, 34, 35ovmpt2d 6953 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗) = ((coe1‘(𝑖𝑀𝑗))‘0))
3736fveq2d 6356 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘(Poly1𝑅))‘(𝑖(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))𝑗)) = ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))
3837mpt2eq3dva 6884 . . 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 2794 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑇‘(𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))))
402, 15m2cpminvid2lem 20761 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
41 simpl2 1230 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑅 ∈ Ring)
42 simprl 811 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
43 simprr 813 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑦𝑁)
4416adantr 472 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1𝑅))))
4510, 11, 12, 42, 43, 44matecld 20434 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥𝑀𝑦) ∈ (Base‘(Poly1𝑅)))
4645, 19, 21sylancl 697 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅))
4715, 25, 8, 11ply1sclcl 19858 . . . . . . . 8 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑥𝑀𝑦))‘0) ∈ (Base‘𝑅)) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1𝑅)))
4841, 46, 47syl2anc 696 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ (Base‘(Poly1𝑅)))
49 eqid 2760 . . . . . . . . 9 (coe1‘((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0))) = (coe1‘((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
5015, 11, 49, 20ply1coe1eq 19870 . . . . . . . 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 213 . . . . . . 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 1477 . . . . . 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 247 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦))
5453ralrimivva 3109 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → ∀𝑥𝑁𝑦𝑁 ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦))
55 eqidd 2761 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))))
56 oveq12 6822 . . . . . . . . . . . 12 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
5756fveq2d 6356 . . . . . . . . . . 11 ((𝑖 = 𝑥𝑗 = 𝑦) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑥𝑀𝑦)))
5857fveq1d 6354 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = 𝑦) → ((coe1‘(𝑖𝑀𝑗))‘0) = ((coe1‘(𝑥𝑀𝑦))‘0))
5958fveq2d 6356 . . . . . . . . 9 ((𝑖 = 𝑥𝑗 = 𝑦) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) = ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
6059adantl 473 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) ∧ (𝑖 = 𝑥𝑗 = 𝑦)) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) = ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
61 simplr 809 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → 𝑥𝑁)
62 simpr 479 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → 𝑦𝑁)
63 fvexd 6364 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) ∈ V)
6455, 60, 61, 62, 63ovmpt2d 6953 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)))
6564eqeq1d 2762 . . . . . 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 682 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ((𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦) ↔ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑥𝑀𝑦))‘0)) = (𝑥𝑀𝑦)))
67662ralbidva 3126 . . . 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 247 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)))𝑦) = (𝑥𝑀𝑦))
69 fvexd 6364 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (Poly1𝑅) ∈ V)
7063ad2ant1 1128 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
71163ad2ant1 1128 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → 𝑀 ∈ (Base‘(𝑁 Mat (Poly1𝑅))))
7210, 11, 12, 33, 34, 71matecld 20434 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Poly1𝑅)))
73 eqid 2760 . . . . . . . 8 (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
7473, 11, 15, 8coe1fvalcl 19784 . . . . . . 7 (((𝑖𝑀𝑗) ∈ (Base‘(Poly1𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅))
7572, 19, 74sylancl 697 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅))
7615, 25, 8, 11ply1sclcl 19858 . . . . . 6 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘0) ∈ (Base‘𝑅)) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) ∈ (Base‘(Poly1𝑅)))
7770, 75, 76syl2anc 696 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0)) ∈ (Base‘(Poly1𝑅)))
7810, 11, 12, 5, 69, 77matbas2d 20431 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) ∈ (Base‘(𝑁 Mat (Poly1𝑅))))
7910, 12eqmat 20432 . . . 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 696 . . 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 247 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘(Poly1𝑅))‘((coe1‘(𝑖𝑀𝑗))‘0))) = 𝑀)
824, 39, 813eqtrd 2798 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑇‘(𝐼𝑀)) = 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cfv 6049  (class class class)co 6813  cmpt2 6815  Fincfn 8121  0cc0 10128  0cn0 11484  Basecbs 16059  Ringcrg 18747  algSccascl 19513  Poly1cpl1 19749  coe1cco1 19750   Mat cmat 20415   ConstPolyMat ccpmat 20710   matToPolyMat cmat2pmat 20711   cPolyMatToMat ccpmat2mat 20712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-ot 4330  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-ofr 7063  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-sup 8513  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-hom 16168  df-cco 16169  df-0g 16304  df-gsum 16305  df-prds 16310  df-pws 16312  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-mulg 17742  df-subg 17792  df-ghm 17859  df-cntz 17950  df-cmn 18395  df-abl 18396  df-mgp 18690  df-ur 18702  df-srg 18706  df-ring 18749  df-subrg 18980  df-lmod 19067  df-lss 19135  df-sra 19374  df-rgmod 19375  df-ascl 19516  df-psr 19558  df-mvr 19559  df-mpl 19560  df-opsr 19562  df-psr1 19752  df-vr1 19753  df-ply1 19754  df-coe1 19755  df-dsmm 20278  df-frlm 20293  df-mat 20416  df-cpmat 20713  df-mat2pmat 20714  df-cpmat2mat 20715
This theorem is referenced by:  m2cpmfo  20763  m2cpminv  20767  cpmadumatpoly  20890  cayhamlem4  20895
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