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Theorem aaitgo 38433
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 3263 . . 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 12005 . . . . 5 ℚ ⊆ ℂ
3 itgoval 38432 . . . . 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 2836 . . 3 (𝑎 ∈ (IntgOver‘ℚ) ↔ 𝑎 ∈ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
6 aacn 24377 . . . . 5 (𝑎 ∈ 𝔸 → 𝑎 ∈ ℂ)
7 mpaacl 38424 . . . . . 6 (𝑎 ∈ 𝔸 → (minPolyAA‘𝑎) ∈ (Poly‘ℚ))
8 mpaaroot 38426 . . . . . 6 (𝑎 ∈ 𝔸 → ((minPolyAA‘𝑎)‘𝑎) = 0)
9 mpaadgr 38425 . . . . . . . 8 (𝑎 ∈ 𝔸 → (deg‘(minPolyAA‘𝑎)) = (degAA𝑎))
109fveq2d 6383 . . . . . . 7 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = ((coeff‘(minPolyAA‘𝑎))‘(degAA𝑎)))
11 mpaamn 38427 . . . . . . 7 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(degAA𝑎)) = 1)
1210, 11eqtrd 2799 . . . . . 6 (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)
13 fveq1 6378 . . . . . . . . 9 (𝑏 = (minPolyAA‘𝑎) → (𝑏𝑎) = ((minPolyAA‘𝑎)‘𝑎))
1413eqeq1d 2767 . . . . . . . 8 (𝑏 = (minPolyAA‘𝑎) → ((𝑏𝑎) = 0 ↔ ((minPolyAA‘𝑎)‘𝑎) = 0))
15 fveq2 6379 . . . . . . . . . 10 (𝑏 = (minPolyAA‘𝑎) → (coeff‘𝑏) = (coeff‘(minPolyAA‘𝑎)))
16 fveq2 6379 . . . . . . . . . 10 (𝑏 = (minPolyAA‘𝑎) → (deg‘𝑏) = (deg‘(minPolyAA‘𝑎)))
1715, 16fveq12d 6386 . . . . . . . . 9 (𝑏 = (minPolyAA‘𝑎) → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))))
1817eqeq1d 2767 . . . . . . . 8 (𝑏 = (minPolyAA‘𝑎) → (((coeff‘𝑏)‘(deg‘𝑏)) = 1 ↔ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1))
1914, 18anbi12d 624 . . . . . . 7 (𝑏 = (minPolyAA‘𝑎) → (((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) ↔ (((minPolyAA‘𝑎)‘𝑎) = 0 ∧ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)))
2019rspcev 3462 . . . . . 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 865 . . . . 5 (𝑎 ∈ 𝔸 → ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1))
226, 21jca 507 . . . 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 24317 . . . . . . . . . . . . . . 15 (coeff‘0𝑝) = (ℕ0 × {0})
2524fveq1i 6380 . . . . . . . . . . . . . 14 ((coeff‘0𝑝)‘(deg‘0𝑝)) = ((ℕ0 × {0})‘(deg‘0𝑝))
26 dgr0 24323 . . . . . . . . . . . . . . . 16 (deg‘0𝑝) = 0
27 0nn0 11559 . . . . . . . . . . . . . . . 16 0 ∈ ℕ0
2826, 27eqeltri 2840 . . . . . . . . . . . . . . 15 (deg‘0𝑝) ∈ ℕ0
29 c0ex 10291 . . . . . . . . . . . . . . . 16 0 ∈ V
3029fvconst2 6666 . . . . . . . . . . . . . . 15 ((deg‘0𝑝) ∈ ℕ0 → ((ℕ0 × {0})‘(deg‘0𝑝)) = 0)
3128, 30ax-mp 5 . . . . . . . . . . . . . 14 ((ℕ0 × {0})‘(deg‘0𝑝)) = 0
3225, 31eqtri 2787 . . . . . . . . . . . . 13 ((coeff‘0𝑝)‘(deg‘0𝑝)) = 0
33 0ne1 11347 . . . . . . . . . . . . 13 0 ≠ 1
3432, 33eqnetri 3007 . . . . . . . . . . . 12 ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1
35 fveq2 6379 . . . . . . . . . . . . . 14 (𝑏 = 0𝑝 → (coeff‘𝑏) = (coeff‘0𝑝))
36 fveq2 6379 . . . . . . . . . . . . . 14 (𝑏 = 0𝑝 → (deg‘𝑏) = (deg‘0𝑝))
3735, 36fveq12d 6386 . . . . . . . . . . . . 13 (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘0𝑝)‘(deg‘0𝑝)))
3837neeq1d 2996 . . . . . . . . . . . 12 (𝑏 = 0𝑝 → (((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1 ↔ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1))
3934, 38mpbiri 249 . . . . . . . . . . 11 (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1)
4039necon2i 2971 . . . . . . . . . 10 (((coeff‘𝑏)‘(deg‘𝑏)) = 1 → 𝑏 ≠ 0𝑝)
4140ad2antll 720 . . . . . . . . 9 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ≠ 0𝑝)
42 eldifsn 4474 . . . . . . . . 9 (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ (𝑏 ∈ (Poly‘ℚ) ∧ 𝑏 ≠ 0𝑝))
4323, 41, 42sylanbrc 578 . . . . . . . 8 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}))
44 simprl 787 . . . . . . . 8 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏𝑎) = 0)
4543, 44jca 507 . . . . . . 7 ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ (𝑏𝑎) = 0))
4645reximi2 3156 . . . . . 6 (∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0)
4746anim2i 610 . . . . 5 ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0))
48 elqaa 24382 . . . . 5 (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏𝑎) = 0))
4947, 48sylibr 225 . . . 4 ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑎 ∈ 𝔸)
5022, 49impbii 200 . . 3 (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
511, 5, 503bitr4ri 295 . 2 (𝑎 ∈ 𝔸 ↔ 𝑎 ∈ (IntgOver‘ℚ))
5251eqriv 2762 1 𝔸 = (IntgOver‘ℚ)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wcel 2155  wne 2937  wrex 3056  {crab 3059  cdif 3731  wss 3734  {csn 4336   × cxp 5277  cfv 6070  cc 10191  0cc0 10193  1c1 10194  0cn0 11542  cq 11994  0𝑝c0p 23741  Polycply 24245  coeffccoe 24247  degcdgr 24248  𝔸caa 24374  degAAcdgraa 38411  minPolyAAcmpaa 38412  IntgOvercitgo 38428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-addf 10272
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-n0 11543  df-z 11629  df-uz 11892  df-q 11995  df-rp 12034  df-fz 12539  df-fzo 12679  df-fl 12806  df-mod 12882  df-seq 13014  df-exp 13073  df-hash 13327  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-clim 14518  df-rlim 14519  df-sum 14716  df-0p 23742  df-ply 24249  df-coe 24251  df-dgr 24252  df-aa 24375  df-dgraa 38413  df-mpaa 38414  df-itgo 38430
This theorem is referenced by: (None)
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