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Theorem chpmatfval 20914
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 20911 . . . 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 6851 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
5 chpmatfval.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
64, 5syl6eqr 2817 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
76fveq2d 6379 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
8 chpmatfval.b . . . . . 6 𝐵 = (Base‘𝐴)
97, 8syl6eqr 2817 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
10 simpl 474 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
11 simpr 477 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑟 = 𝑅)
1211fveq2d 6379 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
13 chpmatfval.p . . . . . . . . 9 𝑃 = (Poly1𝑅)
1412, 13syl6eqr 2817 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = 𝑃)
1510, 14oveq12d 6860 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 maDet (Poly1𝑟)) = (𝑁 maDet 𝑃))
16 chpmatfval.d . . . . . . 7 𝐷 = (𝑁 maDet 𝑃)
1715, 16syl6eqr 2817 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 maDet (Poly1𝑟)) = 𝐷)
18 fveq2 6375 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
1918adantl 473 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
2019, 13syl6eqr 2817 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = 𝑃)
2110, 20oveq12d 6860 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = (𝑁 Mat 𝑃))
22 chpmatfval.y . . . . . . . . . 10 𝑌 = (𝑁 Mat 𝑃)
2321, 22syl6eqr 2817 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = 𝑌)
2423fveq2d 6379 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1𝑟))) = (-g𝑌))
25 chpmatfval.s . . . . . . . 8 = (-g𝑌)
2624, 25syl6eqr 2817 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1𝑟))) = )
2723fveq2d 6379 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1𝑟))) = ( ·𝑠𝑌))
28 chpmatfval.m . . . . . . . . 9 · = ( ·𝑠𝑌)
2927, 28syl6eqr 2817 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1𝑟))) = · )
30 fveq2 6375 . . . . . . . . . 10 (𝑟 = 𝑅 → (var1𝑟) = (var1𝑅))
31 chpmatfval.x . . . . . . . . . 10 𝑋 = (var1𝑅)
3230, 31syl6eqr 2817 . . . . . . . . 9 (𝑟 = 𝑅 → (var1𝑟) = 𝑋)
3332adantl 473 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1𝑟) = 𝑋)
3423fveq2d 6379 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1𝑟))) = (1r𝑌))
35 chpmatfval.i . . . . . . . . 9 1 = (1r𝑌)
3634, 35syl6eqr 2817 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1𝑟))) = 1 )
3729, 33, 36oveq123d 6863 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟)))) = (𝑋 · 1 ))
38 oveq12 6851 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = (𝑁 matToPolyMat 𝑅))
39 chpmatfval.t . . . . . . . . 9 𝑇 = (𝑁 matToPolyMat 𝑅)
4038, 39syl6eqr 2817 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = 𝑇)
4140fveq1d 6377 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑛 matToPolyMat 𝑟)‘𝑚) = (𝑇𝑚))
4226, 37, 41oveq123d 6863 . . . . . 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 6382 . . . . 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 4892 . . . 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 473 . . 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 474 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
47 elex 3365 . . . 4 (𝑅𝑉𝑅 ∈ V)
4847adantl 473 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
498fvexi 6389 . . . 4 𝐵 ∈ V
50 mptexg 6677 . . . 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, 51ovmpt2d 6986 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 CharPlyMat 𝑅) = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
531, 52syl5eq 2811 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  cmpt 4888  cfv 6068  (class class class)co 6842  cmpt2 6844  Fincfn 8160  Basecbs 16130   ·𝑠 cvsca 16218  -gcsg 17691  1rcur 18768  var1cv1 19819  Poly1cpl1 19820   Mat cmat 20489   maDet cmdat 20667   matToPolyMat cmat2pmat 20788   CharPlyMat cchpmat 20910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-chpmat 20911
This theorem is referenced by:  chpmatval  20915
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