Theorem aaitgo 43740

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.

Step	Hyp	Ref	Expression
1		rabid 3436	. . 3 ⊢ (𝑎 ∈ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
2		qsscn 12962	. . . . 5 ⊢ ℚ ⊆ ℂ
3		itgoval 43739	. . . . 5 ⊢ (ℚ ⊆ ℂ → (IntgOver‘ℚ) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
4	2, 3	ax-mp 5	. . . 4 ⊢ (IntgOver‘ℚ) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)}
5	4	eleq2i 2855	. . 3 ⊢ (𝑎 ∈ (IntgOver‘ℚ) ↔ 𝑎 ∈ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
6		aacn 26382	. . . . 5 ⊢ (𝑎 ∈ 𝔸 → 𝑎 ∈ ℂ)
7		mpaacl 43731	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → (minPolyAA‘𝑎) ∈ (Poly‘ℚ))
8		mpaaroot 43733	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → ((minPolyAA‘𝑎)‘𝑎) = 0)
9		mpaadgr 43732	. . . . . . . 8 ⊢ (𝑎 ∈ 𝔸 → (deg‘(minPolyAA‘𝑎)) = (degAA‘𝑎))
10	9	fveq2d 6872	. . . . . . 7 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = ((coeff‘(minPolyAA‘𝑎))‘(degAA‘𝑎)))
11		mpaamn 43734	. . . . . . 7 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(degAA‘𝑎)) = 1)
12	10, 11	eqtrd 2798	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)
13		fveq1 6867	. . . . . . . . 9 ⊢ (𝑏 = (minPolyAA‘𝑎) → (𝑏‘𝑎) = ((minPolyAA‘𝑎)‘𝑎))
14	13	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑏 = (minPolyAA‘𝑎) → ((𝑏‘𝑎) = 0 ↔ ((minPolyAA‘𝑎)‘𝑎) = 0))
15		fveq2 6868	. . . . . . . . . 10 ⊢ (𝑏 = (minPolyAA‘𝑎) → (coeff‘𝑏) = (coeff‘(minPolyAA‘𝑎)))
16		fveq2 6868	. . . . . . . . . 10 ⊢ (𝑏 = (minPolyAA‘𝑎) → (deg‘𝑏) = (deg‘(minPolyAA‘𝑎)))
17	15, 16	fveq12d 6875	. . . . . . . . 9 ⊢ (𝑏 = (minPolyAA‘𝑎) → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))))
18	17	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑏 = (minPolyAA‘𝑎) → (((coeff‘𝑏)‘(deg‘𝑏)) = 1 ↔ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1))
19	14, 18	anbi12d 641	. . . . . . 7 ⊢ (𝑏 = (minPolyAA‘𝑎) → (((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) ↔ (((minPolyAA‘𝑎)‘𝑎) = 0 ∧ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)))
20	19	rspcev 3582	. . . . . 6 ⊢ (((minPolyAA‘𝑎) ∈ (Poly‘ℚ) ∧ (((minPolyAA‘𝑎)‘𝑎) = 0 ∧ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)) → ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1))
21	7, 8, 12, 20	syl12anc 847	. . . . 5 ⊢ (𝑎 ∈ 𝔸 → ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1))
22	6, 21	jca 519	. . . 4 ⊢ (𝑎 ∈ 𝔸 → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
23		simpl 486	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ (Poly‘ℚ))
24		coe0 26317	. . . . . . . . . . . . . . 15 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
25	24	fveq1i 6869	. . . . . . . . . . . . . 14 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) = ((ℕ0 × {0})‘(deg‘0𝑝))
26		dgr0 26323	. . . . . . . . . . . . . . . 16 ⊢ (deg‘0𝑝) = 0
27		0nn0 12497	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ0
28	26, 27	eqeltri 2859	. . . . . . . . . . . . . . 15 ⊢ (deg‘0𝑝) ∈ ℕ0
29		c0ex 11174	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
30	29	fvconst2 7189	. . . . . . . . . . . . . . 15 ⊢ ((deg‘0𝑝) ∈ ℕ0 → ((ℕ0 × {0})‘(deg‘0𝑝)) = 0)
31	28, 30	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((ℕ0 × {0})‘(deg‘0𝑝)) = 0
32	25, 31	eqtri 2786	. . . . . . . . . . . . 13 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) = 0
33		0ne1 12290	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
34	32, 33	eqnetri 3028	. . . . . . . . . . . 12 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1
35		fveq2 6868	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 0𝑝 → (coeff‘𝑏) = (coeff‘0𝑝))
36		fveq2 6868	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 0𝑝 → (deg‘𝑏) = (deg‘0𝑝))
37	35, 36	fveq12d 6875	. . . . . . . . . . . . 13 ⊢ (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘0𝑝)‘(deg‘0𝑝)))
38	37	neeq1d 3017	. . . . . . . . . . . 12 ⊢ (𝑏 = 0𝑝 → (((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1 ↔ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1))
39	34, 38	mpbiri 260	. . . . . . . . . . 11 ⊢ (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1)
40	39	necon2i 2992	. . . . . . . . . 10 ⊢ (((coeff‘𝑏)‘(deg‘𝑏)) = 1 → 𝑏 ≠ 0𝑝)
41	40	ad2antll 739	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ≠ 0𝑝)
42		eldifsn 4747	. . . . . . . . 9 ⊢ (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ (𝑏 ∈ (Poly‘ℚ) ∧ 𝑏 ≠ 0𝑝))
43	23, 41, 42	sylanbrc 592	. . . . . . . 8 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}))
44		simprl 780	. . . . . . . 8 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏‘𝑎) = 0)
45	43, 44	jca 519	. . . . . . 7 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ (𝑏‘𝑎) = 0))
46	45	reximi2 3096	. . . . . 6 ⊢ (∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏‘𝑎) = 0)
47	46	anim2i 626	. . . . 5 ⊢ ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏‘𝑎) = 0))
48		elqaa 26387	. . . . 5 ⊢ (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏‘𝑎) = 0))
49	47, 48	sylibr 236	. . . 4 ⊢ ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑎 ∈ 𝔸)
50	22, 49	impbii 211	. . 3 ⊢ (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
51	1, 5, 50	3bitr4ri 306	. 2 ⊢ (𝑎 ∈ 𝔸 ↔ 𝑎 ∈ (IntgOver‘ℚ))
52	51	eqriv 2760	1 ⊢ 𝔸 = (IntgOver‘ℚ)

Syntax hints:   ∧ wa 399   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3087  {crab 3415   ∖ cdif 3902   ⊆ wss 3905  {csn 4583   × cxp 5646  ‘cfv 6522  ℂcc 11072  0cc0 11074  1c1 11075  ℕ₀cn0 12482  ℚcq 12950  0_𝑝c0p 25732  Polycply 26245  coeffccoe 26247  degcdgr 26248  𝔸caa 26379  deg_AAcdgraa 43718  minPolyAAcmpaa 43719  IntgOvercitgo 43735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-inf2 9597  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151  ax-pre-sup 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-isom 6531  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-of 7661  df-om 7848  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-er 8679  df-map 8811  df-pm 8812  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932  df-sup 9389  df-inf 9390  df-oi 9459  df-card 9898  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-div 11846  df-nn 12212  df-2 12281  df-3 12282  df-n0 12483  df-z 12570  df-uz 12841  df-q 12951  df-rp 12995  df-fz 13514  df-fzo 13661  df-fl 13803  df-mod 13881  df-seq 14016  df-exp 14076  df-hash 14345  df-cj 15127  df-re 15128  df-im 15129  df-sqrt 15263  df-abs 15264  df-clim 15516  df-rlim 15517  df-sum 15715  df-0p 25733  df-ply 26249  df-coe 26251  df-dgr 26252  df-aa 26380  df-dgraa 43720  df-mpaa 43721  df-itgo 43737

This theorem is referenced by: (None)