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Theorem chpmatfval 20454
 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 20451 . . . 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 6558 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
5 chpmatfval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
64, 5syl6eqr 2662 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
76fveq2d 6107 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
8 chpmatfval.b . . . . . 6 𝐵 = (Base‘𝐴)
97, 8syl6eqr 2662 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
10 simpl 472 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
11 simpr 476 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑟 = 𝑅)
1211fveq2d 6107 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
13 chpmatfval.p . . . . . . . . 9 𝑃 = (Poly1𝑅)
1412, 13syl6eqr 2662 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = 𝑃)
1510, 14oveq12d 6567 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 maDet (Poly1𝑟)) = (𝑁 maDet 𝑃))
16 chpmatfval.d . . . . . . 7 𝐷 = (𝑁 maDet 𝑃)
1715, 16syl6eqr 2662 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 maDet (Poly1𝑟)) = 𝐷)
18 fveq2 6103 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
1918adantl 481 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
2019, 13syl6eqr 2662 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = 𝑃)
2110, 20oveq12d 6567 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = (𝑁 Mat 𝑃))
22 chpmatfval.y . . . . . . . . . 10 𝑌 = (𝑁 Mat 𝑃)
2321, 22syl6eqr 2662 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = 𝑌)
2423fveq2d 6107 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1𝑟))) = (-g𝑌))
25 chpmatfval.s . . . . . . . 8 = (-g𝑌)
2624, 25syl6eqr 2662 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1𝑟))) = )
2723fveq2d 6107 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1𝑟))) = ( ·𝑠𝑌))
28 chpmatfval.m . . . . . . . . 9 · = ( ·𝑠𝑌)
2927, 28syl6eqr 2662 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1𝑟))) = · )
30 fveq2 6103 . . . . . . . . . 10 (𝑟 = 𝑅 → (var1𝑟) = (var1𝑅))
31 chpmatfval.x . . . . . . . . . 10 𝑋 = (var1𝑅)
3230, 31syl6eqr 2662 . . . . . . . . 9 (𝑟 = 𝑅 → (var1𝑟) = 𝑋)
3332adantl 481 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1𝑟) = 𝑋)
3423fveq2d 6107 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1𝑟))) = (1r𝑌))
35 chpmatfval.i . . . . . . . . 9 1 = (1r𝑌)
3634, 35syl6eqr 2662 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1𝑟))) = 1 )
3729, 33, 36oveq123d 6570 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟)))) = (𝑋 · 1 ))
38 oveq12 6558 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = (𝑁 matToPolyMat 𝑅))
39 chpmatfval.t . . . . . . . . 9 𝑇 = (𝑁 matToPolyMat 𝑅)
4038, 39syl6eqr 2662 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = 𝑇)
4140fveq1d 6105 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑛 matToPolyMat 𝑟)‘𝑚) = (𝑇𝑚))
4226, 37, 41oveq123d 6570 . . . . . 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 6109 . . . . 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 4663 . . . 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 481 . . 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 472 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
47 elex 3185 . . . 4 (𝑅𝑉𝑅 ∈ V)
4847adantl 481 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
49 fvex 6113 . . . . 5 (Base‘𝐴) ∈ V
508, 49eqeltri 2684 . . . 4 𝐵 ∈ V
51 mptexg 6389 . . . 4 (𝐵 ∈ V → (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))) ∈ V)
5250, 51mp1i 13 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))) ∈ V)
533, 45, 46, 48, 52ovmpt2d 6686 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 CharPlyMat 𝑅) = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
541, 53syl5eq 2656 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  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:  chpmatval  20455
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