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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 22179 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))))) |
13 | 12 | 3adant3 1132 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))))) |
14 | fveq2 6842 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑇‘𝑚) = (𝑇‘𝑀)) | |
15 | 14 | oveq2d 7373 | . . . 4 ⊢ (𝑚 = 𝑀 → ((𝑋 · 1 ) − (𝑇‘𝑚)) = ((𝑋 · 1 ) − (𝑇‘𝑀))) |
16 | 15 | fveq2d 6846 | . . 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 1138 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
19 | fvexd 6857 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))) ∈ V) | |
20 | 13, 17, 18, 19 | fvmptd 6955 | 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 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 Basecbs 17083 ·𝑠 cvsca 17137 -gcsg 18750 1rcur 19913 var1cv1 21547 Poly1cpl1 21548 Mat cmat 21754 maDet cmdat 21933 matToPolyMat cmat2pmat 22053 CharPlyMat cchpmat 22175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-chpmat 22176 |
This theorem is referenced by: chpmatply1 22181 chpmatval2 22182 chpmat0d 22183 chpmat1d 22185 chpdmat 22190 cpmadurid 22216 cpmidgsum2 22228 |
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