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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 20600 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))) |
11 | 10 | 3adant3 1081 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))) |
12 | oveq1 6657 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 decompPMat 𝑘) = (𝑀 decompPMat 𝑘)) | |
13 | 12 | oveq1d 6665 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) = ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) |
14 | 13 | mpteq2dv 4745 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) |
15 | 14 | oveq2d 6666 | . . 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 1063 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
18 | ovexd 6680 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) ∈ V) | |
19 | 11, 16, 17, 18 | fvmptd 6288 | 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 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℕ0cn0 11292 Basecbs 15857 ·𝑠 cvsca 15945 Σg cgsu 16101 .gcmg 17540 mulGrpcmgp 18489 var1cv1 19546 Poly1cpl1 19547 Mat cmat 20213 decompPMat cdecpmat 20567 pMatToMatPoly cpm2mp 20597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-pm2mp 20598 |
This theorem is referenced by: pm2mpcl 20602 pm2mpf1 20604 pm2mpcoe1 20605 idpm2idmp 20606 mp2pm2mp 20616 pm2mpghm 20621 pm2mpmhmlem2 20624 monmat2matmon 20629 |
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