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Theorem mpaaval 27335
 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 5730 . . . . 5 degAA degAA
21eqeq2d 2449 . . . 4 deg degAA deg degAA
3 fveq2 5730 . . . . 5
43eqeq1d 2446 . . . 4
51fveq2d 5734 . . . . 5 coeffdegAA coeffdegAA
65eqeq1d 2446 . . . 4 coeffdegAA coeffdegAA
72, 4, 63anbi123d 1255 . . 3 deg degAA coeffdegAA deg degAA coeffdegAA
87riotabidv 6553 . 2 Polydeg degAA coeffdegAA Polydeg degAA coeffdegAA
9 df-mpaa 27327 . 2 minPolyAA Polydeg degAA coeffdegAA
10 riotaex 6555 . 2 Polydeg degAA coeffdegAA
118, 9, 10fvmpt 5808 1 minPolyAA Polydeg degAA coeffdegAA
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 937   wceq 1653   wcel 1726  cfv 5456  crio 6544  cc0 8992  c1 8993  cq 10576  Polycply 20105  coeffccoe 20107  degcdgr 20108  caa 20233  degAAcdgraa 27324  minPolyAAcmpaa 27325 This theorem is referenced by:  mpaalem  27336 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-riota 6551  df-mpaa 27327
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