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Mirrors > Home > MPE Home > Th. List > chpmatval | Structured version Visualization version GIF version |
Description: The characteristic polynomial of a (square) matrix (expressed with a determinant). (Contributed by AV, 2-Aug-2019.) |
Ref | Expression |
---|---|
chpmatfval.c | ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
chpmatfval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
chpmatfval.b | ⊢ 𝐵 = (Base‘𝐴) |
chpmatfval.p | ⊢ 𝑃 = (Poly1‘𝑅) |
chpmatfval.y | ⊢ 𝑌 = (𝑁 Mat 𝑃) |
chpmatfval.d | ⊢ 𝐷 = (𝑁 maDet 𝑃) |
chpmatfval.s | ⊢ − = (-g‘𝑌) |
chpmatfval.x | ⊢ 𝑋 = (var1‘𝑅) |
chpmatfval.m | ⊢ · = ( ·𝑠 ‘𝑌) |
chpmatfval.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
chpmatfval.i | ⊢ 1 = (1r‘𝑌) |
Ref | Expression |
---|---|
chpmatval | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chpmatfval.c | . . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) | |
2 | chpmatfval.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | chpmatfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
4 | chpmatfval.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | chpmatfval.y | . . . 4 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
6 | chpmatfval.d | . . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑃) | |
7 | chpmatfval.s | . . . 4 ⊢ − = (-g‘𝑌) | |
8 | chpmatfval.x | . . . 4 ⊢ 𝑋 = (var1‘𝑅) | |
9 | chpmatfval.m | . . . 4 ⊢ · = ( ·𝑠 ‘𝑌) | |
10 | chpmatfval.t | . . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
11 | chpmatfval.i | . . . 4 ⊢ 1 = (1r‘𝑌) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | chpmatfval 20454 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))))) |
13 | 12 | 3adant3 1074 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))))) |
14 | fveq2 6103 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑇‘𝑚) = (𝑇‘𝑀)) | |
15 | 14 | oveq2d 6565 | . . . 4 ⊢ (𝑚 = 𝑀 → ((𝑋 · 1 ) − (𝑇‘𝑚)) = ((𝑋 · 1 ) − (𝑇‘𝑀))) |
16 | 15 | fveq2d 6107 | . . 3 ⊢ (𝑚 = 𝑀 → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀)))) |
17 | 16 | adantl 481 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀)))) |
18 | simp3 1056 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
19 | fvex 6113 | . . 3 ⊢ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))) ∈ V | |
20 | 19 | a1i 11 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))) ∈ V) |
21 | 13, 17, 18, 20 | fvmptd 6197 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 Basecbs 15695 ·𝑠 cvsca 15772 -gcsg 17247 1rcur 18324 var1cv1 19367 Poly1cpl1 19368 Mat cmat 20032 maDet cmdat 20209 matToPolyMat cmat2pmat 20328 CharPlyMat cchpmat 20450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-chpmat 20451 |
This theorem is referenced by: chpmatply1 20456 chpmatval2 20457 chpmat0d 20458 chpmat1d 20460 chpdmat 20465 cpmadurid 20491 cpmidgsum2 20503 |
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