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Mirrors > Home > MPE Home > Th. List > pm2mpfval | Structured version Visualization version GIF version |
Description: A polynomial matrix transformed into a polynomial over matrices. (Contributed by AV, 4-Oct-2019.) (Revised by AV, 5-Dec-2019.) |
Ref | Expression |
---|---|
pm2mpval.p | ⊢ 𝑃 = (Poly1‘𝑅) |
pm2mpval.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
pm2mpval.b | ⊢ 𝐵 = (Base‘𝐶) |
pm2mpval.m | ⊢ ∗ = ( ·𝑠 ‘𝑄) |
pm2mpval.e | ⊢ ↑ = (.g‘(mulGrp‘𝑄)) |
pm2mpval.x | ⊢ 𝑋 = (var1‘𝐴) |
pm2mpval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
pm2mpval.q | ⊢ 𝑄 = (Poly1‘𝐴) |
pm2mpval.t | ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) |
Ref | Expression |
---|---|
pm2mpfval | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2mpval.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | pm2mpval.c | . . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
3 | pm2mpval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
4 | pm2mpval.m | . . . 4 ⊢ ∗ = ( ·𝑠 ‘𝑄) | |
5 | pm2mpval.e | . . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑄)) | |
6 | pm2mpval.x | . . . 4 ⊢ 𝑋 = (var1‘𝐴) | |
7 | pm2mpval.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
8 | pm2mpval.q | . . . 4 ⊢ 𝑄 = (Poly1‘𝐴) | |
9 | pm2mpval.t | . . . 4 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | pm2mpval 22144 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))) |
11 | 10 | 3adant3 1132 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))) |
12 | oveq1 7364 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 decompPMat 𝑘) = (𝑀 decompPMat 𝑘)) | |
13 | 12 | oveq1d 7372 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) = ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) |
14 | 13 | mpteq2dv 5207 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) |
15 | 14 | oveq2d 7373 | . . 3 ⊢ (𝑚 = 𝑀 → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))) |
16 | 15 | adantl 482 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))) |
17 | simp3 1138 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
18 | ovexd 7392 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ V) | |
19 | 11, 16, 17, 18 | fvmptd 6955 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℕ0cn0 12413 Basecbs 17083 ·𝑠 cvsca 17137 Σg cgsu 17322 .gcmg 18872 mulGrpcmgp 19896 var1cv1 21547 Poly1cpl1 21548 Mat cmat 21754 decompPMat cdecpmat 22111 pMatToMatPoly cpm2mp 22141 |
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-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-ral 3065 df-rex 3074 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-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 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-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-pm2mp 22142 |
This theorem is referenced by: pm2mpcl 22146 pm2mpf1 22148 pm2mpcoe1 22149 idpm2idmp 22150 mp2pm2mp 22160 pm2mpghm 22165 pm2mpmhmlem2 22168 monmat2matmon 22173 |
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