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Theorem aaitgo 38226
Description: The standard algebraic numbers 𝔸 are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
aaitgo 𝔸 = (IntgOver‘ℚ)

Proof of Theorem aaitgo
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabid 3246 . . 3 (𝑎 ∈ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
2 qsscn 11984 . . . . 5 ℚ ⊆ ℂ
3 itgoval 38225 . . . . 5 (ℚ ⊆ ℂ → (IntgOver‘ℚ) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
42, 3ax-mp 5 . . . 4 (IntgOver‘ℚ) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)}
54eleq2i 2823 . . 3 (𝑎 ∈ (IntgOver‘ℚ) ↔ 𝑎 ∈ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
6 aacn 24263 . . . . 5 (𝑎 ∈ 𝔸 → 𝑎 ∈ ℂ)
7 mpaacl 38217 . . . . . 6 (𝑎 ∈ 𝔸 → (minPolyAA‘𝑎) ∈ (Poly‘ℚ))
8 mpaaroot 38219 . . . . . 6 (𝑎 ∈ 𝔸 → ((minPolyAA‘𝑎)‘𝑎) = 0)
9 mpaadgr 38218 . . . . . . . 8 (𝑎 ∈ 𝔸 → (deg‘(minPolyAA‘𝑎)) = (degAA𝑎))
109fveq2d 6348 . . . . . . 7 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = ((coeff‘(minPolyAA‘𝑎))‘(degAA𝑎)))
11 mpaamn 38220 . . . . . . 7 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(degAA𝑎)) = 1)
1210, 11eqtrd 2786 . . . . . 6 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)
13 fveq1 6343 . . . . . . . . 9 (𝑏 = (minPolyAA‘𝑎) → (𝑏𝑎) = ((minPolyAA‘𝑎)‘𝑎))
1413eqeq1d 2754 . . . . . . . 8 (𝑏 = (minPolyAA‘𝑎) → ((𝑏𝑎) = 0 ↔ ((minPolyAA‘𝑎)‘𝑎) = 0))
15 fveq2 6344 . . . . . . . . . 10 (𝑏 = (minPolyAA‘𝑎) → (coeff‘𝑏) = (coeff‘(minPolyAA‘𝑎)))
16 fveq2 6344 . . . . . . . . . 10 (𝑏 = (minPolyAA‘𝑎) → (deg‘𝑏) = (deg‘(minPolyAA‘𝑎)))
1715, 16fveq12d 6350 . . . . . . . . 9 (𝑏 = (minPolyAA‘𝑎) → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))))
1817eqeq1d 2754 . . . . . . . 8 (𝑏 = (minPolyAA‘𝑎) → (((coeff‘𝑏)‘(deg‘𝑏)) = 1 ↔ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1))
1914, 18anbi12d 749 . . . . . . 7 (𝑏 = (minPolyAA‘𝑎) → (((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) ↔ (((minPolyAA‘𝑎)‘𝑎) = 0 ∧ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)))
2019rspcev 3441 . . . . . 6 (((minPolyAA‘𝑎) ∈ (Poly‘ℚ) ∧ (((minPolyAA‘𝑎)‘𝑎) = 0 ∧ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)) → ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1))
217, 8, 12, 20syl12anc 1471 . . . . 5 (𝑎 ∈ 𝔸 → ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1))
226, 21jca 555 . . . 4 (𝑎 ∈ 𝔸 → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
23 simpl 474 . . . . . . . . 9 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ (Poly‘ℚ))
24 coe0 24203 . . . . . . . . . . . . . . 15 (coeff‘0𝑝) = (ℕ0 × {0})
2524fveq1i 6345 . . . . . . . . . . . . . 14 ((coeff‘0𝑝)‘(deg‘0𝑝)) = ((ℕ0 × {0})‘(deg‘0𝑝))
26 dgr0 24209 . . . . . . . . . . . . . . . 16 (deg‘0𝑝) = 0
27 0nn0 11491 . . . . . . . . . . . . . . . 16 0 ∈ ℕ0
2826, 27eqeltri 2827 . . . . . . . . . . . . . . 15 (deg‘0𝑝) ∈ ℕ0
29 c0ex 10218 . . . . . . . . . . . . . . . 16 0 ∈ V
3029fvconst2 6625 . . . . . . . . . . . . . . 15 ((deg‘0𝑝) ∈ ℕ0 → ((ℕ0 × {0})‘(deg‘0𝑝)) = 0)
3128, 30ax-mp 5 . . . . . . . . . . . . . 14 ((ℕ0 × {0})‘(deg‘0𝑝)) = 0
3225, 31eqtri 2774 . . . . . . . . . . . . 13 ((coeff‘0𝑝)‘(deg‘0𝑝)) = 0
33 0ne1 11272 . . . . . . . . . . . . 13 0 ≠ 1
3432, 33eqnetri 2994 . . . . . . . . . . . 12 ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1
35 fveq2 6344 . . . . . . . . . . . . . 14 (𝑏 = 0𝑝 → (coeff‘𝑏) = (coeff‘0𝑝))
36 fveq2 6344 . . . . . . . . . . . . . 14 (𝑏 = 0𝑝 → (deg‘𝑏) = (deg‘0𝑝))
3735, 36fveq12d 6350 . . . . . . . . . . . . 13 (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘0𝑝)‘(deg‘0𝑝)))
3837neeq1d 2983 . . . . . . . . . . . 12 (𝑏 = 0𝑝 → (((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1 ↔ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1))
3934, 38mpbiri 248 . . . . . . . . . . 11 (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1)
4039necon2i 2958 . . . . . . . . . 10 (((coeff‘𝑏)‘(deg‘𝑏)) = 1 → 𝑏 ≠ 0𝑝)
4140ad2antll 767 . . . . . . . . 9 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ≠ 0𝑝)
42 eldifsn 4454 . . . . . . . . 9 (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ (𝑏 ∈ (Poly‘ℚ) ∧ 𝑏 ≠ 0𝑝))
4323, 41, 42sylanbrc 701 . . . . . . . 8 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}))
44 simprl 811 . . . . . . . 8 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏𝑎) = 0)
4543, 44jca 555 . . . . . . 7 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ (𝑏𝑎) = 0))
4645reximi2 3140 . . . . . 6 (∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0)
4746anim2i 594 . . . . 5 ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0))
48 elqaa 24268 . . . . 5 (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0))
4947, 48sylibr 224 . . . 4 ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑎 ∈ 𝔸)
5022, 49impbii 199 . . 3 (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
511, 5, 503bitr4ri 293 . 2 (𝑎 ∈ 𝔸 ↔ 𝑎 ∈ (IntgOver‘ℚ))
5251eqriv 2749 1 𝔸 = (IntgOver‘ℚ)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1624  wcel 2131  wne 2924  wrex 3043  {crab 3046  cdif 3704  wss 3707  {csn 4313   × cxp 5256  cfv 6041  cc 10118  0cc0 10120  1c1 10121  0cn0 11476  cq 11973  0𝑝c0p 23627  Polycply 24131  coeffccoe 24133  degcdgr 24134  𝔸caa 24260  degAAcdgraa 38204  minPolyAAcmpaa 38205  IntgOvercitgo 38221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198  ax-addf 10199
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-pm 8018  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8505  df-inf 8506  df-oi 8572  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-2 11263  df-3 11264  df-n0 11477  df-z 11562  df-uz 11872  df-q 11974  df-rp 12018  df-fz 12512  df-fzo 12652  df-fl 12779  df-mod 12855  df-seq 12988  df-exp 13047  df-hash 13304  df-cj 14030  df-re 14031  df-im 14032  df-sqrt 14166  df-abs 14167  df-clim 14410  df-rlim 14411  df-sum 14608  df-0p 23628  df-ply 24135  df-coe 24137  df-dgr 24138  df-aa 24261  df-dgraa 38206  df-mpaa 38207  df-itgo 38223
This theorem is referenced by: (None)
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