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Theorem aaitgo 39192
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 3312 . . 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 12173 . . . . 5 ℚ ⊆ ℂ
3 itgoval 39191 . . . . 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 2852 . . 3 (𝑎 ∈ (IntgOver‘ℚ) ↔ 𝑎 ∈ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
6 aacn 24625 . . . . 5 (𝑎 ∈ 𝔸 → 𝑎 ∈ ℂ)
7 mpaacl 39183 . . . . . 6 (𝑎 ∈ 𝔸 → (minPolyAA‘𝑎) ∈ (Poly‘ℚ))
8 mpaaroot 39185 . . . . . 6 (𝑎 ∈ 𝔸 → ((minPolyAA‘𝑎)‘𝑎) = 0)
9 mpaadgr 39184 . . . . . . . 8 (𝑎 ∈ 𝔸 → (deg‘(minPolyAA‘𝑎)) = (degAA𝑎))
109fveq2d 6501 . . . . . . 7 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = ((coeff‘(minPolyAA‘𝑎))‘(degAA𝑎)))
11 mpaamn 39186 . . . . . . 7 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(degAA𝑎)) = 1)
1210, 11eqtrd 2809 . . . . . 6 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)
13 fveq1 6496 . . . . . . . . 9 (𝑏 = (minPolyAA‘𝑎) → (𝑏𝑎) = ((minPolyAA‘𝑎)‘𝑎))
1413eqeq1d 2775 . . . . . . . 8 (𝑏 = (minPolyAA‘𝑎) → ((𝑏𝑎) = 0 ↔ ((minPolyAA‘𝑎)‘𝑎) = 0))
15 fveq2 6497 . . . . . . . . . 10 (𝑏 = (minPolyAA‘𝑎) → (coeff‘𝑏) = (coeff‘(minPolyAA‘𝑎)))
16 fveq2 6497 . . . . . . . . . 10 (𝑏 = (minPolyAA‘𝑎) → (deg‘𝑏) = (deg‘(minPolyAA‘𝑎)))
1715, 16fveq12d 6504 . . . . . . . . 9 (𝑏 = (minPolyAA‘𝑎) → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))))
1817eqeq1d 2775 . . . . . . . 8 (𝑏 = (minPolyAA‘𝑎) → (((coeff‘𝑏)‘(deg‘𝑏)) = 1 ↔ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1))
1914, 18anbi12d 622 . . . . . . 7 (𝑏 = (minPolyAA‘𝑎) → (((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) ↔ (((minPolyAA‘𝑎)‘𝑎) = 0 ∧ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)))
2019rspcev 3530 . . . . . 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 825 . . . . 5 (𝑎 ∈ 𝔸 → ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1))
226, 21jca 504 . . . 4 (𝑎 ∈ 𝔸 → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
23 simpl 475 . . . . . . . . 9 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ (Poly‘ℚ))
24 coe0 24565 . . . . . . . . . . . . . . 15 (coeff‘0𝑝) = (ℕ0 × {0})
2524fveq1i 6498 . . . . . . . . . . . . . 14 ((coeff‘0𝑝)‘(deg‘0𝑝)) = ((ℕ0 × {0})‘(deg‘0𝑝))
26 dgr0 24571 . . . . . . . . . . . . . . . 16 (deg‘0𝑝) = 0
27 0nn0 11723 . . . . . . . . . . . . . . . 16 0 ∈ ℕ0
2826, 27eqeltri 2857 . . . . . . . . . . . . . . 15 (deg‘0𝑝) ∈ ℕ0
29 c0ex 10432 . . . . . . . . . . . . . . . 16 0 ∈ V
3029fvconst2 6792 . . . . . . . . . . . . . . 15 ((deg‘0𝑝) ∈ ℕ0 → ((ℕ0 × {0})‘(deg‘0𝑝)) = 0)
3128, 30ax-mp 5 . . . . . . . . . . . . . 14 ((ℕ0 × {0})‘(deg‘0𝑝)) = 0
3225, 31eqtri 2797 . . . . . . . . . . . . 13 ((coeff‘0𝑝)‘(deg‘0𝑝)) = 0
33 0ne1 11510 . . . . . . . . . . . . 13 0 ≠ 1
3432, 33eqnetri 3032 . . . . . . . . . . . 12 ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1
35 fveq2 6497 . . . . . . . . . . . . . 14 (𝑏 = 0𝑝 → (coeff‘𝑏) = (coeff‘0𝑝))
36 fveq2 6497 . . . . . . . . . . . . . 14 (𝑏 = 0𝑝 → (deg‘𝑏) = (deg‘0𝑝))
3735, 36fveq12d 6504 . . . . . . . . . . . . 13 (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘0𝑝)‘(deg‘0𝑝)))
3837neeq1d 3021 . . . . . . . . . . . 12 (𝑏 = 0𝑝 → (((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1 ↔ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1))
3934, 38mpbiri 250 . . . . . . . . . . 11 (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1)
4039necon2i 2996 . . . . . . . . . 10 (((coeff‘𝑏)‘(deg‘𝑏)) = 1 → 𝑏 ≠ 0𝑝)
4140ad2antll 717 . . . . . . . . 9 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ≠ 0𝑝)
42 eldifsn 4590 . . . . . . . . 9 (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ (𝑏 ∈ (Poly‘ℚ) ∧ 𝑏 ≠ 0𝑝))
4323, 41, 42sylanbrc 575 . . . . . . . 8 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}))
44 simprl 759 . . . . . . . 8 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏𝑎) = 0)
4543, 44jca 504 . . . . . . 7 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ (𝑏𝑎) = 0))
4645reximi2 3186 . . . . . 6 (∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0)
4746anim2i 608 . . . . 5 ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0))
48 elqaa 24630 . . . . 5 (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0))
4947, 48sylibr 226 . . . 4 ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑎 ∈ 𝔸)
5022, 49impbii 201 . . 3 (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
511, 5, 503bitr4ri 296 . 2 (𝑎 ∈ 𝔸 ↔ 𝑎 ∈ (IntgOver‘ℚ))
5251eqriv 2770 1 𝔸 = (IntgOver‘ℚ)
Colors of variables: wff setvar class
Syntax hints:  wa 387   = wceq 1508  wcel 2051  wne 2962  wrex 3084  {crab 3087  cdif 3821  wss 3824  {csn 4436   × cxp 5402  cfv 6186  cc 10332  0cc0 10334  1c1 10335  0cn0 11706  cq 12161  0𝑝c0p 23989  Polycply 24493  coeffccoe 24495  degcdgr 24496  𝔸caa 24622  degAAcdgraa 39170  minPolyAAcmpaa 39171  IntgOvercitgo 39187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-inf2 8897  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411  ax-pre-sup 10412  ax-addf 10413
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-se 5364  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-isom 6195  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-of 7226  df-om 7396  df-1st 7500  df-2nd 7501  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-1o 7904  df-oadd 7908  df-er 8088  df-map 8207  df-pm 8208  df-en 8306  df-dom 8307  df-sdom 8308  df-fin 8309  df-sup 8700  df-inf 8701  df-oi 8768  df-card 9161  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-div 11098  df-nn 11439  df-2 11502  df-3 11503  df-n0 11707  df-z 11793  df-uz 12058  df-q 12162  df-rp 12204  df-fz 12708  df-fzo 12849  df-fl 12976  df-mod 13052  df-seq 13184  df-exp 13244  df-hash 13505  df-cj 14318  df-re 14319  df-im 14320  df-sqrt 14454  df-abs 14455  df-clim 14705  df-rlim 14706  df-sum 14903  df-0p 23990  df-ply 24497  df-coe 24499  df-dgr 24500  df-aa 24623  df-dgraa 39172  df-mpaa 39173  df-itgo 39189
This theorem is referenced by: (None)
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