Theorem aaitgo 43151

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 3427	. . 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 12919	. . . . 5 ⊢ ℚ ⊆ ℂ
3		itgoval 43150	. . . . 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 2820	. . 3 ⊢ (𝑎 ∈ (IntgOver‘ℚ) ↔ 𝑎 ∈ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
6		aacn 26225	. . . . 5 ⊢ (𝑎 ∈ 𝔸 → 𝑎 ∈ ℂ)
7		mpaacl 43142	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → (minPolyAA‘𝑎) ∈ (Poly‘ℚ))
8		mpaaroot 43144	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → ((minPolyAA‘𝑎)‘𝑎) = 0)
9		mpaadgr 43143	. . . . . . . 8 ⊢ (𝑎 ∈ 𝔸 → (deg‘(minPolyAA‘𝑎)) = (degAA‘𝑎))
10	9	fveq2d 6862	. . . . . . 7 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = ((coeff‘(minPolyAA‘𝑎))‘(degAA‘𝑎)))
11		mpaamn 43145	. . . . . . 7 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(degAA‘𝑎)) = 1)
12	10, 11	eqtrd 2764	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)
13		fveq1 6857	. . . . . . . . 9 ⊢ (𝑏 = (minPolyAA‘𝑎) → (𝑏‘𝑎) = ((minPolyAA‘𝑎)‘𝑎))
14	13	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑏 = (minPolyAA‘𝑎) → ((𝑏‘𝑎) = 0 ↔ ((minPolyAA‘𝑎)‘𝑎) = 0))
15		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑏 = (minPolyAA‘𝑎) → (coeff‘𝑏) = (coeff‘(minPolyAA‘𝑎)))
16		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑏 = (minPolyAA‘𝑎) → (deg‘𝑏) = (deg‘(minPolyAA‘𝑎)))
17	15, 16	fveq12d 6865	. . . . . . . . 9 ⊢ (𝑏 = (minPolyAA‘𝑎) → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))))
18	17	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑏 = (minPolyAA‘𝑎) → (((coeff‘𝑏)‘(deg‘𝑏)) = 1 ↔ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1))
19	14, 18	anbi12d 632	. . . . . . 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 3588	. . . . . 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 836	. . . . 5 ⊢ (𝑎 ∈ 𝔸 → ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1))
22	6, 21	jca 511	. . . 4 ⊢ (𝑎 ∈ 𝔸 → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
23		simpl 482	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ (Poly‘ℚ))
24		coe0 26161	. . . . . . . . . . . . . . 15 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
25	24	fveq1i 6859	. . . . . . . . . . . . . 14 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) = ((ℕ0 × {0})‘(deg‘0𝑝))
26		dgr0 26168	. . . . . . . . . . . . . . . 16 ⊢ (deg‘0𝑝) = 0
27		0nn0 12457	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ0
28	26, 27	eqeltri 2824	. . . . . . . . . . . . . . 15 ⊢ (deg‘0𝑝) ∈ ℕ0
29		c0ex 11168	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
30	29	fvconst2 7178	. . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . 13 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) = 0
33		0ne1 12257	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
34	32, 33	eqnetri 2995	. . . . . . . . . . . 12 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1
35		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 0𝑝 → (coeff‘𝑏) = (coeff‘0𝑝))
36		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 0𝑝 → (deg‘𝑏) = (deg‘0𝑝))
37	35, 36	fveq12d 6865	. . . . . . . . . . . . 13 ⊢ (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘0𝑝)‘(deg‘0𝑝)))
38	37	neeq1d 2984	. . . . . . . . . . . 12 ⊢ (𝑏 = 0𝑝 → (((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1 ↔ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1))
39	34, 38	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1)
40	39	necon2i 2959	. . . . . . . . . 10 ⊢ (((coeff‘𝑏)‘(deg‘𝑏)) = 1 → 𝑏 ≠ 0𝑝)
41	40	ad2antll 729	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ≠ 0𝑝)
42		eldifsn 4750	. . . . . . . . 9 ⊢ (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ (𝑏 ∈ (Poly‘ℚ) ∧ 𝑏 ≠ 0𝑝))
43	23, 41, 42	sylanbrc 583	. . . . . . . 8 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}))
44		simprl 770	. . . . . . . 8 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏‘𝑎) = 0)
45	43, 44	jca 511	. . . . . . 7 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ (𝑏‘𝑎) = 0))
46	45	reximi2 3062	. . . . . 6 ⊢ (∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏‘𝑎) = 0)
47	46	anim2i 617	. . . . 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 26230	. . . . 5 ⊢ (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏‘𝑎) = 0))
49	47, 48	sylibr 234	. . . 4 ⊢ ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑎 ∈ 𝔸)
50	22, 49	impbii 209	. . 3 ⊢ (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
51	1, 5, 50	3bitr4ri 304	. 2 ⊢ (𝑎 ∈ 𝔸 ↔ 𝑎 ∈ (IntgOver‘ℚ))
52	51	eqriv 2726	1 ⊢ 𝔸 = (IntgOver‘ℚ)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  {crab 3405   ∖ cdif 3911   ⊆ wss 3914  {csn 4589   × cxp 5636  ‘cfv 6511  ℂcc 11066  0cc0 11068  1c1 11069  ℕ₀cn0 12442  ℚcq 12907  0_𝑝c0p 25570  Polycply 26089  coeffccoe 26091  degcdgr 26092  𝔸caa 26222  deg_AAcdgraa 43129  minPolyAAcmpaa 43130  IntgOvercitgo 43146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-z 12530  df-uz 12794  df-q 12908  df-rp 12952  df-fz 13469  df-fzo 13616  df-fl 13754  df-mod 13832  df-seq 13967  df-exp 14027  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-clim 15454  df-rlim 15455  df-sum 15653  df-0p 25571  df-ply 26093  df-coe 26095  df-dgr 26096  df-aa 26223  df-dgraa 43131  df-mpaa 43132  df-itgo 43148

This theorem is referenced by: (None)