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Theorem mpaaval 35716
 Description: Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
mpaaval minPolyAA Polydeg degAA coeffdegAA
Distinct variable group:   ,

Proof of Theorem mpaaval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5881 . . . . 5 degAA degAA
21eqeq2d 2443 . . . 4 deg degAA deg degAA
3 fveq2 5881 . . . . 5
43eqeq1d 2431 . . . 4
51fveq2d 5885 . . . . 5 coeffdegAA coeffdegAA
65eqeq1d 2431 . . . 4 coeffdegAA coeffdegAA
72, 4, 63anbi123d 1335 . . 3 deg degAA coeffdegAA deg degAA coeffdegAA
87riotabidv 6269 . 2 Polydeg degAA coeffdegAA Polydeg degAA coeffdegAA
9 df-mpaa 35708 . 2 minPolyAA Polydeg degAA coeffdegAA
10 riotaex 6271 . 2 Polydeg degAA coeffdegAA
118, 9, 10fvmpt 5964 1 minPolyAA Polydeg degAA coeffdegAA
 Colors of variables: wff setvar class Syntax hints:   wi 4   w3a 982   wceq 1437   wcel 1870  cfv 5601  crio 6266  cc0 9538  c1 9539  cq 11264  Polycply 23006  coeffccoe 23008  degcdgr 23009  caa 23135  degAAcdgraa 35705  minPolyAAcmpaa 35706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-riota 6267  df-mpaa 35708 This theorem is referenced by:  mpaalem  35717
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