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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 21406 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))) |
11 | 10 | 3adant3 1128 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))) |
12 | oveq1 7166 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 decompPMat 𝑘) = (𝑀 decompPMat 𝑘)) | |
13 | 12 | oveq1d 7174 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) = ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) |
14 | 13 | mpteq2dv 5165 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) |
15 | 14 | oveq2d 7175 | . . 3 ⊢ (𝑚 = 𝑀 → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))) |
16 | 15 | adantl 484 | . 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 1134 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
18 | ovexd 7194 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ V) | |
19 | 11, 16, 17, 18 | fvmptd 6778 | 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 1083 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 Fincfn 8512 ℕ0cn0 11900 Basecbs 16486 ·𝑠 cvsca 16572 Σg cgsu 16717 .gcmg 18227 mulGrpcmgp 19242 var1cv1 20347 Poly1cpl1 20348 Mat cmat 21019 decompPMat cdecpmat 21373 pMatToMatPoly cpm2mp 21403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-pm2mp 21404 |
This theorem is referenced by: pm2mpcl 21408 pm2mpf1 21410 pm2mpcoe1 21411 idpm2idmp 21412 mp2pm2mp 21422 pm2mpghm 21427 pm2mpmhmlem2 21430 monmat2matmon 21435 |
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