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Theorem mpaaval 42606
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 6902 . . . . 5 (π‘Ž = 𝐴 β†’ (degAAβ€˜π‘Ž) = (degAAβ€˜π΄))
21eqeq2d 2739 . . . 4 (π‘Ž = 𝐴 β†’ ((degβ€˜π‘) = (degAAβ€˜π‘Ž) ↔ (degβ€˜π‘) = (degAAβ€˜π΄)))
3 fveqeq2 6911 . . . 4 (π‘Ž = 𝐴 β†’ ((π‘β€˜π‘Ž) = 0 ↔ (π‘β€˜π΄) = 0))
4 2fveq3 6907 . . . . 5 (π‘Ž = 𝐴 β†’ ((coeffβ€˜π‘)β€˜(degAAβ€˜π‘Ž)) = ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)))
54eqeq1d 2730 . . . 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 7384 . 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 42598 . 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 7386 . 2 (℩𝑝 ∈ (Polyβ€˜β„š)((degβ€˜π‘) = (degAAβ€˜π΄) ∧ (π‘β€˜π΄) = 0 ∧ ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)) = 1)) ∈ V
107, 8, 9fvmpt 7010 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 6553  β„©crio 7381  0cc0 11146  1c1 11147  β„šcq 12970  Polycply 26138  coeffccoe 26140  degcdgr 26141  π”Έcaa 26269  degAAcdgraa 42595  minPolyAAcmpaa 42596
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-riota 7382  df-mpaa 42598
This theorem is referenced by:  mpaalem  42607
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