Theorem mpaalem 43095

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)) = (degAA‘𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1)))

Proof of Theorem mpaalem

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpaaval 43094	. . 3 ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)))
2		mpaaeu 43093	. . . 4 ⊢ (𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))
3		riotacl2 7398	. . . 4 ⊢ (∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) → (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)})
4	2, 3	syl 17	. . 3 ⊢ (𝐴 ∈ 𝔸 → (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)})
5	1, 4	eqeltrd 2837	. 2 ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)})
6		fveqeq2 6910	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((deg‘𝑝) = (degAA‘𝐴) ↔ (deg‘(minPolyAA‘𝐴)) = (degAA‘𝐴)))
7		fveq1 6900	. . . . 5 ⊢ (𝑝 = (minPolyAA‘𝐴) → (𝑝‘𝐴) = ((minPolyAA‘𝐴)‘𝐴))
8	7	eqeq1d 2735	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((𝑝‘𝐴) = 0 ↔ ((minPolyAA‘𝐴)‘𝐴) = 0))
9		fveq2 6901	. . . . . 6 ⊢ (𝑝 = (minPolyAA‘𝐴) → (coeff‘𝑝) = (coeff‘(minPolyAA‘𝐴)))
10	9	fveq1d 6903	. . . . 5 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((coeff‘𝑝)‘(degAA‘𝐴)) = ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)))
11	10	eqeq1d 2735	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → (((coeff‘𝑝)‘(degAA‘𝐴)) = 1 ↔ ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1))
12	6, 8, 11	3anbi123d 1434	. . 3 ⊢ (𝑝 = (minPolyAA‘𝐴) → (((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1) ↔ ((deg‘(minPolyAA‘𝐴)) = (degAA‘𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1)))
13	12	elrab 3695	. 2 ⊢ ((minPolyAA‘𝐴) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)} ↔ ((minPolyAA‘𝐴) ∈ (Poly‘ℚ) ∧ ((deg‘(minPolyAA‘𝐴)) = (degAA‘𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1)))
14	5, 13	sylib 218	1 ⊢ (𝐴 ∈ 𝔸 → ((minPolyAA‘𝐴) ∈ (Poly‘ℚ) ∧ ((deg‘(minPolyAA‘𝐴)) = (degAA‘𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1535   ∈ wcel 2104  ∃!wreu 3374  {crab 3432  ‘cfv 6558  ℩crio 7380  0cc0 11146  1c1 11147  ℚcq 12981  Polycply 26219  coeffccoe 26221  degcdgr 26222  𝔸caa 26352  deg_AAcdgraa 43083  minPolyAAcmpaa 43084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5366  ax-pr 5430  ax-un 7747  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-int 4954  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  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 6317  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7381  df-ov 7428  df-oprab 7429  df-mpo 7430  df-of 7691  df-om 7881  df-1st 8007  df-2nd 8008  df-frecs 8299  df-wrecs 8330  df-recs 8404  df-rdg 8443  df-1o 8499  df-er 8738  df-map 8861  df-pm 8862  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-sup 9473  df-inf 9474  df-oi 9541  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11485  df-neg 11486  df-div 11912  df-nn 12258  df-2 12320  df-3 12321  df-n0 12518  df-z 12605  df-uz 12870  df-q 12982  df-rp 13026  df-fz 13538  df-fzo 13682  df-fl 13818  df-mod 13896  df-seq 14029  df-exp 14089  df-hash 14356  df-cj 15124  df-re 15125  df-im 15126  df-sqrt 15260  df-abs 15261  df-clim 15510  df-rlim 15511  df-sum 15709  df-0p 25700  df-ply 26223  df-coe 26225  df-dgr 26226  df-aa 26353  df-dgraa 43085  df-mpaa 43086

This theorem is referenced by:  mpaacl  43096  mpaadgr  43097  mpaaroot  43098  mpaamn  43099