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Theorem chpmatfval 21127
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 21124 . . . 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 7030 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
5 chpmatfval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
64, 5syl6eqr 2849 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
76fveq2d 6547 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
8 chpmatfval.b . . . . . 6 𝐵 = (Base‘𝐴)
97, 8syl6eqr 2849 . . . . 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 6547 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
13 chpmatfval.p . . . . . . . . 9 𝑃 = (Poly1𝑅)
1412, 13syl6eqr 2849 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = 𝑃)
1510, 14oveq12d 7039 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 maDet (Poly1𝑟)) = (𝑁 maDet 𝑃))
16 chpmatfval.d . . . . . . 7 𝐷 = (𝑁 maDet 𝑃)
1715, 16syl6eqr 2849 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 maDet (Poly1𝑟)) = 𝐷)
18 fveq2 6543 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
1918adantl 482 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
2019, 13syl6eqr 2849 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = 𝑃)
2110, 20oveq12d 7039 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = (𝑁 Mat 𝑃))
22 chpmatfval.y . . . . . . . . . 10 𝑌 = (𝑁 Mat 𝑃)
2321, 22syl6eqr 2849 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = 𝑌)
2423fveq2d 6547 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1𝑟))) = (-g𝑌))
25 chpmatfval.s . . . . . . . 8 = (-g𝑌)
2624, 25syl6eqr 2849 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1𝑟))) = )
2723fveq2d 6547 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1𝑟))) = ( ·𝑠𝑌))
28 chpmatfval.m . . . . . . . . 9 · = ( ·𝑠𝑌)
2927, 28syl6eqr 2849 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1𝑟))) = · )
30 fveq2 6543 . . . . . . . . . 10 (𝑟 = 𝑅 → (var1𝑟) = (var1𝑅))
31 chpmatfval.x . . . . . . . . . 10 𝑋 = (var1𝑅)
3230, 31syl6eqr 2849 . . . . . . . . 9 (𝑟 = 𝑅 → (var1𝑟) = 𝑋)
3332adantl 482 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1𝑟) = 𝑋)
3423fveq2d 6547 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1𝑟))) = (1r𝑌))
35 chpmatfval.i . . . . . . . . 9 1 = (1r𝑌)
3634, 35syl6eqr 2849 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1𝑟))) = 1 )
3729, 33, 36oveq123d 7042 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟)))) = (𝑋 · 1 ))
38 oveq12 7030 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = (𝑁 matToPolyMat 𝑅))
39 chpmatfval.t . . . . . . . . 9 𝑇 = (𝑁 matToPolyMat 𝑅)
4038, 39syl6eqr 2849 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = 𝑇)
4140fveq1d 6545 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑛 matToPolyMat 𝑟)‘𝑚) = (𝑇𝑚))
4226, 37, 41oveq123d 7042 . . . . . 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 6550 . . . . 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 5050 . . . 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 3455 . . . 4 (𝑅𝑉𝑅 ∈ V)
4847adantl 482 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
498fvexi 6557 . . . 4 𝐵 ∈ V
50 mptexg 6855 . . . 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 7163 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 CharPlyMat 𝑅) = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
531, 52syl5eq 2843 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  Vcvv 3437  cmpt 5045  cfv 6230  (class class class)co 7021  cmpo 7023  Fincfn 8362  Basecbs 16317   ·𝑠 cvsca 16403  -gcsg 17868  1rcur 18946  var1cv1 20032  Poly1cpl1 20033   Mat cmat 20705   maDet cmdat 20882   matToPolyMat cmat2pmat 21001   CharPlyMat cchpmat 21123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pr 5226
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-ov 7024  df-oprab 7025  df-mpo 7026  df-chpmat 21124
This theorem is referenced by:  chpmatval  21128
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