Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  chpmatfval Structured version   Visualization version   GIF version

Theorem chpmatfval 21530
 Description: Value of the characteristic polynomial function. (Contributed by AV, 2-Aug-2019.)
Hypotheses
Ref Expression
chpmatfval.c 𝐶 = (𝑁 CharPlyMat 𝑅)
chpmatfval.a 𝐴 = (𝑁 Mat 𝑅)
chpmatfval.b 𝐵 = (Base‘𝐴)
chpmatfval.p 𝑃 = (Poly1𝑅)
chpmatfval.y 𝑌 = (𝑁 Mat 𝑃)
chpmatfval.d 𝐷 = (𝑁 maDet 𝑃)
chpmatfval.s = (-g𝑌)
chpmatfval.x 𝑋 = (var1𝑅)
chpmatfval.m · = ( ·𝑠𝑌)
chpmatfval.t 𝑇 = (𝑁 matToPolyMat 𝑅)
chpmatfval.i 1 = (1r𝑌)
Assertion
Ref Expression
chpmatfval ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
Distinct variable groups:   𝐵,𝑚   𝐷,𝑚   1 ,𝑚   𝑚,𝑁   𝑅,𝑚   𝑚,𝑋   𝑇,𝑚   · ,𝑚   ,𝑚
Allowed substitution hints:   𝐴(𝑚)   𝐶(𝑚)   𝑃(𝑚)   𝑉(𝑚)   𝑌(𝑚)

Proof of Theorem chpmatfval
Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 chpmatfval.c . 2 𝐶 = (𝑁 CharPlyMat 𝑅)
2 df-chpmat 21527 . . . 4 CharPlyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1𝑟))‘(((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟))))(-g‘(𝑛 Mat (Poly1𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))))
32a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → CharPlyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1𝑟))‘(((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟))))(-g‘(𝑛 Mat (Poly1𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚))))))
4 oveq12 7159 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
5 chpmatfval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
64, 5eqtr4di 2811 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
76fveq2d 6662 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
8 chpmatfval.b . . . . . 6 𝐵 = (Base‘𝐴)
97, 8eqtr4di 2811 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
10 simpl 486 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
11 simpr 488 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑟 = 𝑅)
1211fveq2d 6662 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
13 chpmatfval.p . . . . . . . . 9 𝑃 = (Poly1𝑅)
1412, 13eqtr4di 2811 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = 𝑃)
1510, 14oveq12d 7168 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 maDet (Poly1𝑟)) = (𝑁 maDet 𝑃))
16 chpmatfval.d . . . . . . 7 𝐷 = (𝑁 maDet 𝑃)
1715, 16eqtr4di 2811 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 maDet (Poly1𝑟)) = 𝐷)
18 fveq2 6658 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
1918adantl 485 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
2019, 13eqtr4di 2811 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = 𝑃)
2110, 20oveq12d 7168 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = (𝑁 Mat 𝑃))
22 chpmatfval.y . . . . . . . . . 10 𝑌 = (𝑁 Mat 𝑃)
2321, 22eqtr4di 2811 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = 𝑌)
2423fveq2d 6662 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1𝑟))) = (-g𝑌))
25 chpmatfval.s . . . . . . . 8 = (-g𝑌)
2624, 25eqtr4di 2811 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1𝑟))) = )
2723fveq2d 6662 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1𝑟))) = ( ·𝑠𝑌))
28 chpmatfval.m . . . . . . . . 9 · = ( ·𝑠𝑌)
2927, 28eqtr4di 2811 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1𝑟))) = · )
30 fveq2 6658 . . . . . . . . . 10 (𝑟 = 𝑅 → (var1𝑟) = (var1𝑅))
31 chpmatfval.x . . . . . . . . . 10 𝑋 = (var1𝑅)
3230, 31eqtr4di 2811 . . . . . . . . 9 (𝑟 = 𝑅 → (var1𝑟) = 𝑋)
3332adantl 485 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1𝑟) = 𝑋)
3423fveq2d 6662 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1𝑟))) = (1r𝑌))
35 chpmatfval.i . . . . . . . . 9 1 = (1r𝑌)
3634, 35eqtr4di 2811 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1𝑟))) = 1 )
3729, 33, 36oveq123d 7171 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟)))) = (𝑋 · 1 ))
38 oveq12 7159 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = (𝑁 matToPolyMat 𝑅))
39 chpmatfval.t . . . . . . . . 9 𝑇 = (𝑁 matToPolyMat 𝑅)
4038, 39eqtr4di 2811 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = 𝑇)
4140fveq1d 6660 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑛 matToPolyMat 𝑟)‘𝑚) = (𝑇𝑚))
4226, 37, 41oveq123d 7171 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟))))(-g‘(𝑛 Mat (Poly1𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)) = ((𝑋 · 1 ) (𝑇𝑚)))
4317, 42fveq12d 6665 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑛 maDet (Poly1𝑟))‘(((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟))))(-g‘(𝑛 Mat (Poly1𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚))) = (𝐷‘((𝑋 · 1 ) (𝑇𝑚))))
449, 43mpteq12dv 5117 . . . 4 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1𝑟))‘(((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟))))(-g‘(𝑛 Mat (Poly1𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))) = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
4544adantl 485 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1𝑟))‘(((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟))))(-g‘(𝑛 Mat (Poly1𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))) = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
46 simpl 486 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
47 elex 3428 . . . 4 (𝑅𝑉𝑅 ∈ V)
4847adantl 485 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
498fvexi 6672 . . . 4 𝐵 ∈ V
50 mptexg 6975 . . . 4 (𝐵 ∈ V → (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))) ∈ V)
5149, 50mp1i 13 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))) ∈ V)
523, 45, 46, 48, 51ovmpod 7297 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 CharPlyMat 𝑅) = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
531, 52syl5eq 2805 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ↦ cmpt 5112  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152  Fincfn 8527  Basecbs 16541   ·𝑠 cvsca 16627  -gcsg 18171  1rcur 19319  var1cv1 20900  Poly1cpl1 20901   Mat cmat 21107   maDet cmdat 21284   matToPolyMat cmat2pmat 21404   CharPlyMat cchpmat 21526 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-chpmat 21527 This theorem is referenced by:  chpmatval  21531
 Copyright terms: Public domain W3C validator