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Theorem minplyval 33749
Description: Expand the value of the minimal polynomial (𝑀𝐴) for a given element 𝐴. It is defined as the unique monic polynomial of minimal degree which annihilates 𝐴. By ply1annig1p 33748, that polynomial generates the ideal of the annihilators of 𝐴. (Contributed by Thierry Arnoux, 9-Feb-2025.)
Hypotheses
Ref Expression
ply1annig1p.o 𝑂 = (𝐸 evalSub1 𝐹)
ply1annig1p.p 𝑃 = (Poly1‘(𝐸s 𝐹))
ply1annig1p.b 𝐵 = (Base‘𝐸)
ply1annig1p.e (𝜑𝐸 ∈ Field)
ply1annig1p.f (𝜑𝐹 ∈ (SubDRing‘𝐸))
ply1annig1p.a (𝜑𝐴𝐵)
ply1annig1p.0 0 = (0g𝐸)
ply1annig1p.q 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = 0 }
ply1annig1p.k 𝐾 = (RSpan‘𝑃)
ply1annig1p.g 𝐺 = (idlGen1p‘(𝐸s 𝐹))
minplyval.1 𝑀 = (𝐸 minPoly 𝐹)
Assertion
Ref Expression
minplyval (𝜑 → (𝑀𝐴) = (𝐺𝑄))
Distinct variable groups:   0 ,𝑞   𝐴,𝑞   𝑂,𝑞   𝑃,𝑞   𝜑,𝑞   𝐸,𝑞   𝐹,𝑞
Allowed substitution hints:   𝐵(𝑞)   𝑄(𝑞)   𝐺(𝑞)   𝐾(𝑞)   𝑀(𝑞)

Proof of Theorem minplyval
Dummy variables 𝑒 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 minplyval.1 . . 3 𝑀 = (𝐸 minPoly 𝐹)
2 ply1annig1p.e . . . . 5 (𝜑𝐸 ∈ Field)
32elexd 3503 . . . 4 (𝜑𝐸 ∈ V)
4 ply1annig1p.f . . . . 5 (𝜑𝐹 ∈ (SubDRing‘𝐸))
54elexd 3503 . . . 4 (𝜑𝐹 ∈ V)
6 ply1annig1p.b . . . . . . 7 𝐵 = (Base‘𝐸)
76fvexi 6919 . . . . . 6 𝐵 ∈ V
87a1i 11 . . . . 5 (𝜑𝐵 ∈ V)
98mptexd 7245 . . . 4 (𝜑 → (𝑥𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 })) ∈ V)
10 fveq2 6905 . . . . . . . 8 (𝑒 = 𝐸 → (Base‘𝑒) = (Base‘𝐸))
1110, 6eqtr4di 2794 . . . . . . 7 (𝑒 = 𝐸 → (Base‘𝑒) = 𝐵)
1211adantr 480 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → (Base‘𝑒) = 𝐵)
13 oveq12 7441 . . . . . . . . 9 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒s 𝑓) = (𝐸s 𝐹))
1413fveq2d 6909 . . . . . . . 8 ((𝑒 = 𝐸𝑓 = 𝐹) → (idlGen1p‘(𝑒s 𝑓)) = (idlGen1p‘(𝐸s 𝐹)))
15 ply1annig1p.g . . . . . . . 8 𝐺 = (idlGen1p‘(𝐸s 𝐹))
1614, 15eqtr4di 2794 . . . . . . 7 ((𝑒 = 𝐸𝑓 = 𝐹) → (idlGen1p‘(𝑒s 𝑓)) = 𝐺)
17 oveq12 7441 . . . . . . . . . 10 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = (𝐸 evalSub1 𝐹))
18 ply1annig1p.o . . . . . . . . . 10 𝑂 = (𝐸 evalSub1 𝐹)
1917, 18eqtr4di 2794 . . . . . . . . 9 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = 𝑂)
2019dmeqd 5915 . . . . . . . 8 ((𝑒 = 𝐸𝑓 = 𝐹) → dom (𝑒 evalSub1 𝑓) = dom 𝑂)
2119fveq1d 6907 . . . . . . . . . 10 ((𝑒 = 𝐸𝑓 = 𝐹) → ((𝑒 evalSub1 𝑓)‘𝑞) = (𝑂𝑞))
2221fveq1d 6907 . . . . . . . . 9 ((𝑒 = 𝐸𝑓 = 𝐹) → (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = ((𝑂𝑞)‘𝑥))
23 fveq2 6905 . . . . . . . . . . 11 (𝑒 = 𝐸 → (0g𝑒) = (0g𝐸))
2423adantr 480 . . . . . . . . . 10 ((𝑒 = 𝐸𝑓 = 𝐹) → (0g𝑒) = (0g𝐸))
25 ply1annig1p.0 . . . . . . . . . 10 0 = (0g𝐸)
2624, 25eqtr4di 2794 . . . . . . . . 9 ((𝑒 = 𝐸𝑓 = 𝐹) → (0g𝑒) = 0 )
2722, 26eqeq12d 2752 . . . . . . . 8 ((𝑒 = 𝐸𝑓 = 𝐹) → ((((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g𝑒) ↔ ((𝑂𝑞)‘𝑥) = 0 ))
2820, 27rabeqbidv 3454 . . . . . . 7 ((𝑒 = 𝐸𝑓 = 𝐹) → {𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g𝑒)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 })
2916, 28fveq12d 6912 . . . . . 6 ((𝑒 = 𝐸𝑓 = 𝐹) → ((idlGen1p‘(𝑒s 𝑓))‘{𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g𝑒)}) = (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 }))
3012, 29mpteq12dv 5232 . . . . 5 ((𝑒 = 𝐸𝑓 = 𝐹) → (𝑥 ∈ (Base‘𝑒) ↦ ((idlGen1p‘(𝑒s 𝑓))‘{𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g𝑒)})) = (𝑥𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 })))
31 df-minply 33744 . . . . 5 minPoly = (𝑒 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑒) ↦ ((idlGen1p‘(𝑒s 𝑓))‘{𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g𝑒)})))
3230, 31ovmpoga 7588 . . . 4 ((𝐸 ∈ V ∧ 𝐹 ∈ V ∧ (𝑥𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 })) ∈ V) → (𝐸 minPoly 𝐹) = (𝑥𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 })))
333, 5, 9, 32syl3anc 1372 . . 3 (𝜑 → (𝐸 minPoly 𝐹) = (𝑥𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 })))
341, 33eqtrid 2788 . 2 (𝜑𝑀 = (𝑥𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 })))
35 fveqeq2 6914 . . . . . 6 (𝑥 = 𝐴 → (((𝑂𝑞)‘𝑥) = 0 ↔ ((𝑂𝑞)‘𝐴) = 0 ))
3635rabbidv 3443 . . . . 5 (𝑥 = 𝐴 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 } = {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = 0 })
37 ply1annig1p.q . . . . 5 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝐴) = 0 }
3836, 37eqtr4di 2794 . . . 4 (𝑥 = 𝐴 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 } = 𝑄)
3938fveq2d 6909 . . 3 (𝑥 = 𝐴 → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 }) = (𝐺𝑄))
4039adantl 481 . 2 ((𝜑𝑥 = 𝐴) → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂𝑞)‘𝑥) = 0 }) = (𝐺𝑄))
41 ply1annig1p.a . 2 (𝜑𝐴𝐵)
42 fvexd 6920 . 2 (𝜑 → (𝐺𝑄) ∈ V)
4334, 40, 41, 42fvmptd 7022 1 (𝜑 → (𝑀𝐴) = (𝐺𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {crab 3435  Vcvv 3479  cmpt 5224  dom cdm 5684  cfv 6560  (class class class)co 7432  Basecbs 17248  s cress 17275  0gc0g 17485  Fieldcfield 20731  SubDRingcsdrg 20788  RSpancrsp 21218  Poly1cpl1 22179   evalSub1 ces1 22318  idlGen1pcig1p 26170   minPoly cminply 33743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-minply 33744
This theorem is referenced by:  minplycl  33750  minplymindeg  33752  minplyann  33753  minplyirredlem  33754  minplyirred  33755  irngnminplynz  33756  minplym1p  33757  irredminply  33758  algextdeglem4  33762  algextdeglem5  33763
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