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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 21331 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))) |
11 | 10 | 3adant3 1124 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))) |
12 | oveq1 7152 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 decompPMat 𝑘) = (𝑀 decompPMat 𝑘)) | |
13 | 12 | oveq1d 7160 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) = ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) |
14 | 13 | mpteq2dv 5153 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) |
15 | 14 | oveq2d 7161 | . . 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 1130 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
18 | ovexd 7180 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ V) | |
19 | 11, 16, 17, 18 | fvmptd 6767 | 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 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 ℕ0cn0 11885 Basecbs 16471 ·𝑠 cvsca 16557 Σg cgsu 16702 .gcmg 18162 mulGrpcmgp 19168 var1cv1 20272 Poly1cpl1 20273 Mat cmat 20944 decompPMat cdecpmat 21298 pMatToMatPoly cpm2mp 21328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-pm2mp 21329 |
This theorem is referenced by: pm2mpcl 21333 pm2mpf1 21335 pm2mpcoe1 21336 idpm2idmp 21337 mp2pm2mp 21347 pm2mpghm 21352 pm2mpmhmlem2 21355 monmat2matmon 21360 |
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