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Theorem aaitgo 42823
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 3440 . . 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 12996 . . . . 5 ℚ ⊆ ℂ
3 itgoval 42822 . . . . 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 2818 . . 3 (𝑎 ∈ (IntgOver‘ℚ) ↔ 𝑎 ∈ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
6 aacn 26345 . . . . 5 (𝑎 ∈ 𝔸 → 𝑎 ∈ ℂ)
7 mpaacl 42814 . . . . . 6 (𝑎 ∈ 𝔸 → (minPolyAA‘𝑎) ∈ (Poly‘ℚ))
8 mpaaroot 42816 . . . . . 6 (𝑎 ∈ 𝔸 → ((minPolyAA‘𝑎)‘𝑎) = 0)
9 mpaadgr 42815 . . . . . . . 8 (𝑎 ∈ 𝔸 → (deg‘(minPolyAA‘𝑎)) = (degAA𝑎))
109fveq2d 6905 . . . . . . 7 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = ((coeff‘(minPolyAA‘𝑎))‘(degAA𝑎)))
11 mpaamn 42817 . . . . . . 7 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(degAA𝑎)) = 1)
1210, 11eqtrd 2766 . . . . . 6 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)
13 fveq1 6900 . . . . . . . . 9 (𝑏 = (minPolyAA‘𝑎) → (𝑏𝑎) = ((minPolyAA‘𝑎)‘𝑎))
1413eqeq1d 2728 . . . . . . . 8 (𝑏 = (minPolyAA‘𝑎) → ((𝑏𝑎) = 0 ↔ ((minPolyAA‘𝑎)‘𝑎) = 0))
15 fveq2 6901 . . . . . . . . . 10 (𝑏 = (minPolyAA‘𝑎) → (coeff‘𝑏) = (coeff‘(minPolyAA‘𝑎)))
16 fveq2 6901 . . . . . . . . . 10 (𝑏 = (minPolyAA‘𝑎) → (deg‘𝑏) = (deg‘(minPolyAA‘𝑎)))
1715, 16fveq12d 6908 . . . . . . . . 9 (𝑏 = (minPolyAA‘𝑎) → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))))
1817eqeq1d 2728 . . . . . . . 8 (𝑏 = (minPolyAA‘𝑎) → (((coeff‘𝑏)‘(deg‘𝑏)) = 1 ↔ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1))
1914, 18anbi12d 630 . . . . . . 7 (𝑏 = (minPolyAA‘𝑎) → (((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) ↔ (((minPolyAA‘𝑎)‘𝑎) = 0 ∧ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)))
2019rspcev 3608 . . . . . 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 510 . . . 4 (𝑎 ∈ 𝔸 → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
23 simpl 481 . . . . . . . . 9 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ (Poly‘ℚ))
24 coe0 26283 . . . . . . . . . . . . . . 15 (coeff‘0𝑝) = (ℕ0 × {0})
2524fveq1i 6902 . . . . . . . . . . . . . 14 ((coeff‘0𝑝)‘(deg‘0𝑝)) = ((ℕ0 × {0})‘(deg‘0𝑝))
26 dgr0 26290 . . . . . . . . . . . . . . . 16 (deg‘0𝑝) = 0
27 0nn0 12539 . . . . . . . . . . . . . . . 16 0 ∈ ℕ0
2826, 27eqeltri 2822 . . . . . . . . . . . . . . 15 (deg‘0𝑝) ∈ ℕ0
29 c0ex 11258 . . . . . . . . . . . . . . . 16 0 ∈ V
3029fvconst2 7221 . . . . . . . . . . . . . . 15 ((deg‘0𝑝) ∈ ℕ0 → ((ℕ0 × {0})‘(deg‘0𝑝)) = 0)
3128, 30ax-mp 5 . . . . . . . . . . . . . 14 ((ℕ0 × {0})‘(deg‘0𝑝)) = 0
3225, 31eqtri 2754 . . . . . . . . . . . . 13 ((coeff‘0𝑝)‘(deg‘0𝑝)) = 0
33 0ne1 12335 . . . . . . . . . . . . 13 0 ≠ 1
3432, 33eqnetri 3001 . . . . . . . . . . . 12 ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1
35 fveq2 6901 . . . . . . . . . . . . . 14 (𝑏 = 0𝑝 → (coeff‘𝑏) = (coeff‘0𝑝))
36 fveq2 6901 . . . . . . . . . . . . . 14 (𝑏 = 0𝑝 → (deg‘𝑏) = (deg‘0𝑝))
3735, 36fveq12d 6908 . . . . . . . . . . . . 13 (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘0𝑝)‘(deg‘0𝑝)))
3837neeq1d 2990 . . . . . . . . . . . 12 (𝑏 = 0𝑝 → (((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1 ↔ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1))
3934, 38mpbiri 257 . . . . . . . . . . 11 (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1)
4039necon2i 2965 . . . . . . . . . 10 (((coeff‘𝑏)‘(deg‘𝑏)) = 1 → 𝑏 ≠ 0𝑝)
4140ad2antll 727 . . . . . . . . 9 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ≠ 0𝑝)
42 eldifsn 4795 . . . . . . . . 9 (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ (𝑏 ∈ (Poly‘ℚ) ∧ 𝑏 ≠ 0𝑝))
4323, 41, 42sylanbrc 581 . . . . . . . 8 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}))
44 simprl 769 . . . . . . . 8 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏𝑎) = 0)
4543, 44jca 510 . . . . . . 7 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ (𝑏𝑎) = 0))
4645reximi2 3069 . . . . . 6 (∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0)
4746anim2i 615 . . . . 5 ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0))
48 elqaa 26350 . . . . 5 (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0))
4947, 48sylibr 233 . . . 4 ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑎 ∈ 𝔸)
5022, 49impbii 208 . . 3 (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
511, 5, 503bitr4ri 303 . 2 (𝑎 ∈ 𝔸 ↔ 𝑎 ∈ (IntgOver‘ℚ))
5251eqriv 2723 1 𝔸 = (IntgOver‘ℚ)
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1534  wcel 2099  wne 2930  wrex 3060  {crab 3419  cdif 3944  wss 3947  {csn 4633   × cxp 5680  cfv 6554  cc 11156  0cc0 11158  1c1 11159  0cn0 12524  cq 12984  0𝑝c0p 25689  Polycply 26211  coeffccoe 26213  degcdgr 26214  𝔸caa 26342  degAAcdgraa 42801  minPolyAAcmpaa 42802  IntgOvercitgo 42818
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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-pm 8858  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-inf 9486  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-q 12985  df-rp 13029  df-fz 13539  df-fzo 13682  df-fl 13812  df-mod 13890  df-seq 14022  df-exp 14082  df-hash 14348  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-clim 15490  df-rlim 15491  df-sum 15691  df-0p 25690  df-ply 26215  df-coe 26217  df-dgr 26218  df-aa 26343  df-dgraa 42803  df-mpaa 42804  df-itgo 42820
This theorem is referenced by: (None)
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