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Theorem mpaaval 41893
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 6892 . . . . 5 (π‘Ž = 𝐴 β†’ (degAAβ€˜π‘Ž) = (degAAβ€˜π΄))
21eqeq2d 2744 . . . 4 (π‘Ž = 𝐴 β†’ ((degβ€˜π‘) = (degAAβ€˜π‘Ž) ↔ (degβ€˜π‘) = (degAAβ€˜π΄)))
3 fveqeq2 6901 . . . 4 (π‘Ž = 𝐴 β†’ ((π‘β€˜π‘Ž) = 0 ↔ (π‘β€˜π΄) = 0))
4 2fveq3 6897 . . . . 5 (π‘Ž = 𝐴 β†’ ((coeffβ€˜π‘)β€˜(degAAβ€˜π‘Ž)) = ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)))
54eqeq1d 2735 . . . 4 (π‘Ž = 𝐴 β†’ (((coeffβ€˜π‘)β€˜(degAAβ€˜π‘Ž)) = 1 ↔ ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)) = 1))
62, 3, 53anbi123d 1437 . . 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 7367 . 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 41885 . 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 7369 . 2 (℩𝑝 ∈ (Polyβ€˜β„š)((degβ€˜π‘) = (degAAβ€˜π΄) ∧ (π‘β€˜π΄) = 0 ∧ ((coeffβ€˜π‘)β€˜(degAAβ€˜π΄)) = 1)) ∈ V
107, 8, 9fvmpt 6999 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 1088   = wceq 1542   ∈ wcel 2107  β€˜cfv 6544  β„©crio 7364  0cc0 11110  1c1 11111  β„šcq 12932  Polycply 25698  coeffccoe 25700  degcdgr 25701  π”Έcaa 25827  degAAcdgraa 41882  minPolyAAcmpaa 41883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-riota 7365  df-mpaa 41885
This theorem is referenced by:  mpaalem  41894
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