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Theorem minplyval 33433
Description: Expand the value of the minimal polynomial (π‘€β€˜π΄) for a given element 𝐴. It is defined as the unique monic polynomial of minimal degree which annihilates 𝐴. By ply1annig1p 33432, 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 = (Poly1β€˜(𝐸 β†Ύs 𝐹))
ply1annig1p.b 𝐡 = (Baseβ€˜πΈ)
ply1annig1p.e (πœ‘ β†’ 𝐸 ∈ Field)
ply1annig1p.f (πœ‘ β†’ 𝐹 ∈ (SubDRingβ€˜πΈ))
ply1annig1p.a (πœ‘ β†’ 𝐴 ∈ 𝐡)
ply1annig1p.0 0 = (0gβ€˜πΈ)
ply1annig1p.q 𝑄 = {π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π΄) = 0 }
ply1annig1p.k 𝐾 = (RSpanβ€˜π‘ƒ)
ply1annig1p.g 𝐺 = (idlGen1pβ€˜(𝐸 β†Ύs 𝐹))
minplyval.1 𝑀 = (𝐸 minPoly 𝐹)
Assertion
Ref Expression
minplyval (πœ‘ β†’ (π‘€β€˜π΄) = (πΊβ€˜π‘„))
Distinct variable groups:   0 ,π‘ž   𝐴,π‘ž   𝑂,π‘ž   𝑃,π‘ž   πœ‘,π‘ž   𝐸,π‘ž   𝐹,π‘ž
Allowed substitution hints:   𝐡(π‘ž)   𝑄(π‘ž)   𝐺(π‘ž)   𝐾(π‘ž)   𝑀(π‘ž)

Proof of Theorem minplyval
Dummy variables 𝑒 𝑓 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 minplyval.1 . . 3 𝑀 = (𝐸 minPoly 𝐹)
2 ply1annig1p.e . . . . 5 (πœ‘ β†’ 𝐸 ∈ Field)
32elexd 3484 . . . 4 (πœ‘ β†’ 𝐸 ∈ V)
4 ply1annig1p.f . . . . 5 (πœ‘ β†’ 𝐹 ∈ (SubDRingβ€˜πΈ))
54elexd 3484 . . . 4 (πœ‘ β†’ 𝐹 ∈ V)
6 ply1annig1p.b . . . . . . 7 𝐡 = (Baseβ€˜πΈ)
76fvexi 6906 . . . . . 6 𝐡 ∈ V
87a1i 11 . . . . 5 (πœ‘ β†’ 𝐡 ∈ V)
98mptexd 7232 . . . 4 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ (πΊβ€˜{π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 })) ∈ V)
10 fveq2 6892 . . . . . . . 8 (𝑒 = 𝐸 β†’ (Baseβ€˜π‘’) = (Baseβ€˜πΈ))
1110, 6eqtr4di 2783 . . . . . . 7 (𝑒 = 𝐸 β†’ (Baseβ€˜π‘’) = 𝐡)
1211adantr 479 . . . . . 6 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (Baseβ€˜π‘’) = 𝐡)
13 oveq12 7425 . . . . . . . . 9 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (𝑒 β†Ύs 𝑓) = (𝐸 β†Ύs 𝐹))
1413fveq2d 6896 . . . . . . . 8 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (idlGen1pβ€˜(𝑒 β†Ύs 𝑓)) = (idlGen1pβ€˜(𝐸 β†Ύs 𝐹)))
15 ply1annig1p.g . . . . . . . 8 𝐺 = (idlGen1pβ€˜(𝐸 β†Ύs 𝐹))
1614, 15eqtr4di 2783 . . . . . . 7 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (idlGen1pβ€˜(𝑒 β†Ύs 𝑓)) = 𝐺)
17 oveq12 7425 . . . . . . . . . 10 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (𝑒 evalSub1 𝑓) = (𝐸 evalSub1 𝐹))
18 ply1annig1p.o . . . . . . . . . 10 𝑂 = (𝐸 evalSub1 𝐹)
1917, 18eqtr4di 2783 . . . . . . . . 9 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (𝑒 evalSub1 𝑓) = 𝑂)
2019dmeqd 5902 . . . . . . . 8 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ dom (𝑒 evalSub1 𝑓) = dom 𝑂)
2119fveq1d 6894 . . . . . . . . . 10 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ ((𝑒 evalSub1 𝑓)β€˜π‘ž) = (π‘‚β€˜π‘ž))
2221fveq1d 6894 . . . . . . . . 9 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (((𝑒 evalSub1 𝑓)β€˜π‘ž)β€˜π‘₯) = ((π‘‚β€˜π‘ž)β€˜π‘₯))
23 fveq2 6892 . . . . . . . . . . 11 (𝑒 = 𝐸 β†’ (0gβ€˜π‘’) = (0gβ€˜πΈ))
2423adantr 479 . . . . . . . . . 10 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (0gβ€˜π‘’) = (0gβ€˜πΈ))
25 ply1annig1p.0 . . . . . . . . . 10 0 = (0gβ€˜πΈ)
2624, 25eqtr4di 2783 . . . . . . . . 9 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (0gβ€˜π‘’) = 0 )
2722, 26eqeq12d 2741 . . . . . . . 8 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ ((((𝑒 evalSub1 𝑓)β€˜π‘ž)β€˜π‘₯) = (0gβ€˜π‘’) ↔ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 ))
2820, 27rabeqbidv 3437 . . . . . . 7 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ {π‘ž ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)β€˜π‘ž)β€˜π‘₯) = (0gβ€˜π‘’)} = {π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 })
2916, 28fveq12d 6899 . . . . . 6 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ ((idlGen1pβ€˜(𝑒 β†Ύs 𝑓))β€˜{π‘ž ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)β€˜π‘ž)β€˜π‘₯) = (0gβ€˜π‘’)}) = (πΊβ€˜{π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 }))
3012, 29mpteq12dv 5234 . . . . 5 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (π‘₯ ∈ (Baseβ€˜π‘’) ↦ ((idlGen1pβ€˜(𝑒 β†Ύs 𝑓))β€˜{π‘ž ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)β€˜π‘ž)β€˜π‘₯) = (0gβ€˜π‘’)})) = (π‘₯ ∈ 𝐡 ↦ (πΊβ€˜{π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 })))
31 df-minply 33428 . . . . 5 minPoly = (𝑒 ∈ V, 𝑓 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘’) ↦ ((idlGen1pβ€˜(𝑒 β†Ύs 𝑓))β€˜{π‘ž ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)β€˜π‘ž)β€˜π‘₯) = (0gβ€˜π‘’)})))
3230, 31ovmpoga 7572 . . . 4 ((𝐸 ∈ V ∧ 𝐹 ∈ V ∧ (π‘₯ ∈ 𝐡 ↦ (πΊβ€˜{π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 })) ∈ V) β†’ (𝐸 minPoly 𝐹) = (π‘₯ ∈ 𝐡 ↦ (πΊβ€˜{π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 })))
333, 5, 9, 32syl3anc 1368 . . 3 (πœ‘ β†’ (𝐸 minPoly 𝐹) = (π‘₯ ∈ 𝐡 ↦ (πΊβ€˜{π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 })))
341, 33eqtrid 2777 . 2 (πœ‘ β†’ 𝑀 = (π‘₯ ∈ 𝐡 ↦ (πΊβ€˜{π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 })))
35 fveqeq2 6901 . . . . . 6 (π‘₯ = 𝐴 β†’ (((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 ↔ ((π‘‚β€˜π‘ž)β€˜π΄) = 0 ))
3635rabbidv 3427 . . . . 5 (π‘₯ = 𝐴 β†’ {π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 } = {π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π΄) = 0 })
37 ply1annig1p.q . . . . 5 𝑄 = {π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π΄) = 0 }
3836, 37eqtr4di 2783 . . . 4 (π‘₯ = 𝐴 β†’ {π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 } = 𝑄)
3938fveq2d 6896 . . 3 (π‘₯ = 𝐴 β†’ (πΊβ€˜{π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 }) = (πΊβ€˜π‘„))
4039adantl 480 . 2 ((πœ‘ ∧ π‘₯ = 𝐴) β†’ (πΊβ€˜{π‘ž ∈ dom 𝑂 ∣ ((π‘‚β€˜π‘ž)β€˜π‘₯) = 0 }) = (πΊβ€˜π‘„))
41 ply1annig1p.a . 2 (πœ‘ β†’ 𝐴 ∈ 𝐡)
42 fvexd 6907 . 2 (πœ‘ β†’ (πΊβ€˜π‘„) ∈ V)
4334, 40, 41, 42fvmptd 7007 1 (πœ‘ β†’ (π‘€β€˜π΄) = (πΊβ€˜π‘„))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3419  Vcvv 3463   ↦ cmpt 5226  dom cdm 5672  β€˜cfv 6543  (class class class)co 7416  Basecbs 17179   β†Ύs cress 17208  0gc0g 17420  Fieldcfield 20629  SubDRingcsdrg 20678  RSpancrsp 21107  Poly1cpl1 22104   evalSub1 ces1 22241  idlGen1pcig1p 26083   minPoly cminply 33427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-minply 33428
This theorem is referenced by:  minplycl  33434  minplyann  33436  minplyirredlem  33437  minplyirred  33438  irngnminplynz  33439  minplym1p  33440  irredminply  33441  algextdeglem4  33445  algextdeglem5  33446
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