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Theorem chpmatfval 22179
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 22176 . . . 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 7366 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
5 chpmatfval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
64, 5eqtr4di 2794 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
76fveq2d 6846 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
8 chpmatfval.b . . . . . 6 𝐵 = (Base‘𝐴)
97, 8eqtr4di 2794 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
10 simpl 483 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
11 simpr 485 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑟 = 𝑅)
1211fveq2d 6846 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
13 chpmatfval.p . . . . . . . . 9 𝑃 = (Poly1𝑅)
1412, 13eqtr4di 2794 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = 𝑃)
1510, 14oveq12d 7375 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 maDet (Poly1𝑟)) = (𝑁 maDet 𝑃))
16 chpmatfval.d . . . . . . 7 𝐷 = (𝑁 maDet 𝑃)
1715, 16eqtr4di 2794 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 maDet (Poly1𝑟)) = 𝐷)
18 fveq2 6842 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
1918adantl 482 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
2019, 13eqtr4di 2794 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = 𝑃)
2110, 20oveq12d 7375 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = (𝑁 Mat 𝑃))
22 chpmatfval.y . . . . . . . . . 10 𝑌 = (𝑁 Mat 𝑃)
2321, 22eqtr4di 2794 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = 𝑌)
2423fveq2d 6846 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1𝑟))) = (-g𝑌))
25 chpmatfval.s . . . . . . . 8 = (-g𝑌)
2624, 25eqtr4di 2794 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1𝑟))) = )
2723fveq2d 6846 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1𝑟))) = ( ·𝑠𝑌))
28 chpmatfval.m . . . . . . . . 9 · = ( ·𝑠𝑌)
2927, 28eqtr4di 2794 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1𝑟))) = · )
30 fveq2 6842 . . . . . . . . . 10 (𝑟 = 𝑅 → (var1𝑟) = (var1𝑅))
31 chpmatfval.x . . . . . . . . . 10 𝑋 = (var1𝑅)
3230, 31eqtr4di 2794 . . . . . . . . 9 (𝑟 = 𝑅 → (var1𝑟) = 𝑋)
3332adantl 482 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1𝑟) = 𝑋)
3423fveq2d 6846 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1𝑟))) = (1r𝑌))
35 chpmatfval.i . . . . . . . . 9 1 = (1r𝑌)
3634, 35eqtr4di 2794 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1𝑟))) = 1 )
3729, 33, 36oveq123d 7378 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟)))) = (𝑋 · 1 ))
38 oveq12 7366 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = (𝑁 matToPolyMat 𝑅))
39 chpmatfval.t . . . . . . . . 9 𝑇 = (𝑁 matToPolyMat 𝑅)
4038, 39eqtr4di 2794 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = 𝑇)
4140fveq1d 6844 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑛 matToPolyMat 𝑟)‘𝑚) = (𝑇𝑚))
4226, 37, 41oveq123d 7378 . . . . . 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 6849 . . . . 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 5196 . . . 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 482 . . 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 483 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
47 elex 3463 . . . 4 (𝑅𝑉𝑅 ∈ V)
4847adantl 482 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
498fvexi 6856 . . . 4 𝐵 ∈ V
50 mptexg 7171 . . . 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 7507 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 CharPlyMat 𝑅) = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
531, 52eqtrid 2788 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cmpt 5188  cfv 6496  (class class class)co 7357  cmpo 7359  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:  chpmatval  22180
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