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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 20635 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))))) |
13 | 12 | 3adant3 1081 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))))) |
14 | fveq2 6191 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑇‘𝑚) = (𝑇‘𝑀)) | |
15 | 14 | oveq2d 6666 | . . . 4 ⊢ (𝑚 = 𝑀 → ((𝑋 · 1 ) − (𝑇‘𝑚)) = ((𝑋 · 1 ) − (𝑇‘𝑀))) |
16 | 15 | fveq2d 6195 | . . 3 ⊢ (𝑚 = 𝑀 → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀)))) |
17 | 16 | adantl 482 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀)))) |
18 | simp3 1063 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
19 | fvexd 6203 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))) ∈ V) | |
20 | 13, 17, 18, 19 | fvmptd 6288 | 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 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 Basecbs 15857 ·𝑠 cvsca 15945 -gcsg 17424 1rcur 18501 var1cv1 19546 Poly1cpl1 19547 Mat cmat 20213 maDet cmdat 20390 matToPolyMat cmat2pmat 20509 CharPlyMat cchpmat 20631 |
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-chpmat 20632 |
This theorem is referenced by: chpmatply1 20637 chpmatval2 20638 chpmat0d 20639 chpmat1d 20641 chpdmat 20646 cpmadurid 20672 cpmidgsum2 20684 |
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