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Theorem mpaaval 42453
Description: Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
mpaaval (𝐴 ∈ 𝔸 β†’ (minPolyAAβ€˜π΄) = (℩𝑝 ∈ (Polyβ€˜β„š)((degβ€˜π‘) = (degAAβ€˜π΄) ∧ (π‘β€˜π΄) = 0 ∧ ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)) = 1)))
Distinct variable group:   𝐴,𝑝

Proof of Theorem mpaaval
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 fveq2 6884 . . . . 5 (π‘Ž = 𝐴 β†’ (degAAβ€˜π‘Ž) = (degAAβ€˜π΄))
21eqeq2d 2737 . . . 4 (π‘Ž = 𝐴 β†’ ((degβ€˜π‘) = (degAAβ€˜π‘Ž) ↔ (degβ€˜π‘) = (degAAβ€˜π΄)))
3 fveqeq2 6893 . . . 4 (π‘Ž = 𝐴 β†’ ((π‘β€˜π‘Ž) = 0 ↔ (π‘β€˜π΄) = 0))
4 2fveq3 6889 . . . . 5 (π‘Ž = 𝐴 β†’ ((coeffβ€˜π‘)β€˜(degAAβ€˜π‘Ž)) = ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)))
54eqeq1d 2728 . . . 4 (π‘Ž = 𝐴 β†’ (((coeffβ€˜π‘)β€˜(degAAβ€˜π‘Ž)) = 1 ↔ ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)) = 1))
62, 3, 53anbi123d 1432 . . 3 (π‘Ž = 𝐴 β†’ (((degβ€˜π‘) = (degAAβ€˜π‘Ž) ∧ (π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degAAβ€˜π‘Ž)) = 1) ↔ ((degβ€˜π‘) = (degAAβ€˜π΄) ∧ (π‘β€˜π΄) = 0 ∧ ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)) = 1)))
76riotabidv 7362 . 2 (π‘Ž = 𝐴 β†’ (℩𝑝 ∈ (Polyβ€˜β„š)((degβ€˜π‘) = (degAAβ€˜π‘Ž) ∧ (π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degAAβ€˜π‘Ž)) = 1)) = (℩𝑝 ∈ (Polyβ€˜β„š)((degβ€˜π‘) = (degAAβ€˜π΄) ∧ (π‘β€˜π΄) = 0 ∧ ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)) = 1)))
8 df-mpaa 42445 . 2 minPolyAA = (π‘Ž ∈ 𝔸 ↦ (℩𝑝 ∈ (Polyβ€˜β„š)((degβ€˜π‘) = (degAAβ€˜π‘Ž) ∧ (π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degAAβ€˜π‘Ž)) = 1)))
9 riotaex 7364 . 2 (℩𝑝 ∈ (Polyβ€˜β„š)((degβ€˜π‘) = (degAAβ€˜π΄) ∧ (π‘β€˜π΄) = 0 ∧ ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)) = 1)) ∈ V
107, 8, 9fvmpt 6991 1 (𝐴 ∈ 𝔸 β†’ (minPolyAAβ€˜π΄) = (℩𝑝 ∈ (Polyβ€˜β„š)((degβ€˜π‘) = (degAAβ€˜π΄) ∧ (π‘β€˜π΄) = 0 ∧ ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)) = 1)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  β€˜cfv 6536  β„©crio 7359  0cc0 11109  1c1 11110  β„šcq 12933  Polycply 26068  coeffccoe 26070  degcdgr 26071  π”Έcaa 26199  degAAcdgraa 42442  minPolyAAcmpaa 42443
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-riota 7360  df-mpaa 42445
This theorem is referenced by:  mpaalem  42454
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