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Theorem aaitgo 40121
 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 3331 . . 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 12349 . . . . 5 ℚ ⊆ ℂ
3 itgoval 40120 . . . . 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 2881 . . 3 (𝑎 ∈ (IntgOver‘ℚ) ↔ 𝑎 ∈ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
6 aacn 24920 . . . . 5 (𝑎 ∈ 𝔸 → 𝑎 ∈ ℂ)
7 mpaacl 40112 . . . . . 6 (𝑎 ∈ 𝔸 → (minPolyAA‘𝑎) ∈ (Poly‘ℚ))
8 mpaaroot 40114 . . . . . 6 (𝑎 ∈ 𝔸 → ((minPolyAA‘𝑎)‘𝑎) = 0)
9 mpaadgr 40113 . . . . . . . 8 (𝑎 ∈ 𝔸 → (deg‘(minPolyAA‘𝑎)) = (degAA𝑎))
109fveq2d 6649 . . . . . . 7 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = ((coeff‘(minPolyAA‘𝑎))‘(degAA𝑎)))
11 mpaamn 40115 . . . . . . 7 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(degAA𝑎)) = 1)
1210, 11eqtrd 2833 . . . . . 6 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)
13 fveq1 6644 . . . . . . . . 9 (𝑏 = (minPolyAA‘𝑎) → (𝑏𝑎) = ((minPolyAA‘𝑎)‘𝑎))
1413eqeq1d 2800 . . . . . . . 8 (𝑏 = (minPolyAA‘𝑎) → ((𝑏𝑎) = 0 ↔ ((minPolyAA‘𝑎)‘𝑎) = 0))
15 fveq2 6645 . . . . . . . . . 10 (𝑏 = (minPolyAA‘𝑎) → (coeff‘𝑏) = (coeff‘(minPolyAA‘𝑎)))
16 fveq2 6645 . . . . . . . . . 10 (𝑏 = (minPolyAA‘𝑎) → (deg‘𝑏) = (deg‘(minPolyAA‘𝑎)))
1715, 16fveq12d 6652 . . . . . . . . 9 (𝑏 = (minPolyAA‘𝑎) → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))))
1817eqeq1d 2800 . . . . . . . 8 (𝑏 = (minPolyAA‘𝑎) → (((coeff‘𝑏)‘(deg‘𝑏)) = 1 ↔ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1))
1914, 18anbi12d 633 . . . . . . 7 (𝑏 = (minPolyAA‘𝑎) → (((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) ↔ (((minPolyAA‘𝑎)‘𝑎) = 0 ∧ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)))
2019rspcev 3571 . . . . . 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 835 . . . . 5 (𝑎 ∈ 𝔸 → ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1))
226, 21jca 515 . . . 4 (𝑎 ∈ 𝔸 → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
23 simpl 486 . . . . . . . . 9 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ (Poly‘ℚ))
24 coe0 24860 . . . . . . . . . . . . . . 15 (coeff‘0𝑝) = (ℕ0 × {0})
2524fveq1i 6646 . . . . . . . . . . . . . 14 ((coeff‘0𝑝)‘(deg‘0𝑝)) = ((ℕ0 × {0})‘(deg‘0𝑝))
26 dgr0 24866 . . . . . . . . . . . . . . . 16 (deg‘0𝑝) = 0
27 0nn0 11902 . . . . . . . . . . . . . . . 16 0 ∈ ℕ0
2826, 27eqeltri 2886 . . . . . . . . . . . . . . 15 (deg‘0𝑝) ∈ ℕ0
29 c0ex 10626 . . . . . . . . . . . . . . . 16 0 ∈ V
3029fvconst2 6943 . . . . . . . . . . . . . . 15 ((deg‘0𝑝) ∈ ℕ0 → ((ℕ0 × {0})‘(deg‘0𝑝)) = 0)
3128, 30ax-mp 5 . . . . . . . . . . . . . 14 ((ℕ0 × {0})‘(deg‘0𝑝)) = 0
3225, 31eqtri 2821 . . . . . . . . . . . . 13 ((coeff‘0𝑝)‘(deg‘0𝑝)) = 0
33 0ne1 11698 . . . . . . . . . . . . 13 0 ≠ 1
3432, 33eqnetri 3057 . . . . . . . . . . . 12 ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1
35 fveq2 6645 . . . . . . . . . . . . . 14 (𝑏 = 0𝑝 → (coeff‘𝑏) = (coeff‘0𝑝))
36 fveq2 6645 . . . . . . . . . . . . . 14 (𝑏 = 0𝑝 → (deg‘𝑏) = (deg‘0𝑝))
3735, 36fveq12d 6652 . . . . . . . . . . . . 13 (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘0𝑝)‘(deg‘0𝑝)))
3837neeq1d 3046 . . . . . . . . . . . 12 (𝑏 = 0𝑝 → (((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1 ↔ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1))
3934, 38mpbiri 261 . . . . . . . . . . 11 (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1)
4039necon2i 3021 . . . . . . . . . 10 (((coeff‘𝑏)‘(deg‘𝑏)) = 1 → 𝑏 ≠ 0𝑝)
4140ad2antll 728 . . . . . . . . 9 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ≠ 0𝑝)
42 eldifsn 4680 . . . . . . . . 9 (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ (𝑏 ∈ (Poly‘ℚ) ∧ 𝑏 ≠ 0𝑝))
4323, 41, 42sylanbrc 586 . . . . . . . 8 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}))
44 simprl 770 . . . . . . . 8 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏𝑎) = 0)
4543, 44jca 515 . . . . . . 7 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ (𝑏𝑎) = 0))
4645reximi2 3207 . . . . . 6 (∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0)
4746anim2i 619 . . . . 5 ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0))
48 elqaa 24925 . . . . 5 (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0))
4947, 48sylibr 237 . . . 4 ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑎 ∈ 𝔸)
5022, 49impbii 212 . . 3 (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
511, 5, 503bitr4ri 307 . 2 (𝑎 ∈ 𝔸 ↔ 𝑎 ∈ (IntgOver‘ℚ))
5251eqriv 2795 1 𝔸 = (IntgOver‘ℚ)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  {crab 3110   ∖ cdif 3878   ⊆ wss 3881  {csn 4525   × cxp 5517  ‘cfv 6324  ℂcc 10526  0cc0 10528  1c1 10529  ℕ0cn0 11887  ℚcq 12338  0𝑝c0p 24280  Polycply 24788  coeffccoe 24790  degcdgr 24791  𝔸caa 24917  degAAcdgraa 40099  minPolyAAcmpaa 40100  IntgOvercitgo 40116 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-inf2 9090  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605  ax-pre-sup 10606  ax-addf 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7390  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-er 8274  df-map 8393  df-pm 8394  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-sup 8892  df-inf 8893  df-oi 8960  df-card 9354  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-div 11289  df-nn 11628  df-2 11690  df-3 11691  df-n0 11888  df-z 11972  df-uz 12234  df-q 12339  df-rp 12380  df-fz 12888  df-fzo 13031  df-fl 13159  df-mod 13235  df-seq 13367  df-exp 13428  df-hash 13689  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-rlim 14840  df-sum 15037  df-0p 24281  df-ply 24792  df-coe 24794  df-dgr 24795  df-aa 24918  df-dgraa 40101  df-mpaa 40102  df-itgo 40118 This theorem is referenced by: (None)
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