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Theorem mpaalem 43142
Description: Properties of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
mpaalem (𝐴 ∈ 𝔸 → ((minPolyAA‘𝐴) ∈ (Poly‘ℚ) ∧ ((deg‘(minPolyAA‘𝐴)) = (degAA𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(degAA𝐴)) = 1)))

Proof of Theorem mpaalem
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 mpaaval 43141 . . 3 (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)))
2 mpaaeu 43140 . . . 4 (𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
3 riotacl2 7402 . . . 4 (∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) → (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)})
42, 3syl 17 . . 3 (𝐴 ∈ 𝔸 → (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)})
51, 4eqeltrd 2840 . 2 (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)})
6 fveqeq2 6913 . . . 4 (𝑝 = (minPolyAA‘𝐴) → ((deg‘𝑝) = (degAA𝐴) ↔ (deg‘(minPolyAA‘𝐴)) = (degAA𝐴)))
7 fveq1 6903 . . . . 5 (𝑝 = (minPolyAA‘𝐴) → (𝑝𝐴) = ((minPolyAA‘𝐴)‘𝐴))
87eqeq1d 2738 . . . 4 (𝑝 = (minPolyAA‘𝐴) → ((𝑝𝐴) = 0 ↔ ((minPolyAA‘𝐴)‘𝐴) = 0))
9 fveq2 6904 . . . . . 6 (𝑝 = (minPolyAA‘𝐴) → (coeff‘𝑝) = (coeff‘(minPolyAA‘𝐴)))
109fveq1d 6906 . . . . 5 (𝑝 = (minPolyAA‘𝐴) → ((coeff‘𝑝)‘(degAA𝐴)) = ((coeff‘(minPolyAA‘𝐴))‘(degAA𝐴)))
1110eqeq1d 2738 . . . 4 (𝑝 = (minPolyAA‘𝐴) → (((coeff‘𝑝)‘(degAA𝐴)) = 1 ↔ ((coeff‘(minPolyAA‘𝐴))‘(degAA𝐴)) = 1))
126, 8, 113anbi123d 1438 . . 3 (𝑝 = (minPolyAA‘𝐴) → (((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1) ↔ ((deg‘(minPolyAA‘𝐴)) = (degAA𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(degAA𝐴)) = 1)))
1312elrab 3691 . 2 ((minPolyAA‘𝐴) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)} ↔ ((minPolyAA‘𝐴) ∈ (Poly‘ℚ) ∧ ((deg‘(minPolyAA‘𝐴)) = (degAA𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(degAA𝐴)) = 1)))
145, 13sylib 218 1 (𝐴 ∈ 𝔸 → ((minPolyAA‘𝐴) ∈ (Poly‘ℚ) ∧ ((deg‘(minPolyAA‘𝐴)) = (degAA𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(degAA𝐴)) = 1)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  ∃!wreu 3377  {crab 3435  cfv 6559  crio 7385  0cc0 11151  1c1 11152  cq 12986  Polycply 26213  coeffccoe 26215  degcdgr 26216  𝔸caa 26346  degAAcdgraa 43130  minPolyAAcmpaa 43131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751  ax-inf2 9677  ax-cnex 11207  ax-resscn 11208  ax-1cn 11209  ax-icn 11210  ax-addcl 11211  ax-addrcl 11212  ax-mulcl 11213  ax-mulrcl 11214  ax-mulcom 11215  ax-addass 11216  ax-mulass 11217  ax-distr 11218  ax-i2m1 11219  ax-1ne0 11220  ax-1rid 11221  ax-rnegex 11222  ax-rrecex 11223  ax-cnre 11224  ax-pre-lttri 11225  ax-pre-lttrn 11226  ax-pre-ltadd 11227  ax-pre-mulgt0 11228  ax-pre-sup 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-int 4945  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-se 5636  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-isom 6568  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-of 7694  df-om 7884  df-1st 8010  df-2nd 8011  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-1o 8502  df-er 8741  df-map 8864  df-pm 8865  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-sup 9478  df-inf 9479  df-oi 9546  df-card 9975  df-pnf 11293  df-mnf 11294  df-xr 11295  df-ltxr 11296  df-le 11297  df-sub 11490  df-neg 11491  df-div 11917  df-nn 12263  df-2 12325  df-3 12326  df-n0 12523  df-z 12610  df-uz 12875  df-q 12987  df-rp 13031  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-0p 25695  df-ply 26217  df-coe 26219  df-dgr 26220  df-aa 26347  df-dgraa 43132  df-mpaa 43133
This theorem is referenced by:  mpaacl  43143  mpaadgr  43144  mpaaroot  43145  mpaamn  43146
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