Theorem aaitgo 43909

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 3437	. . 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 12979	. . . . 5 ⊢ ℚ ⊆ ℂ
3		itgoval 43908	. . . . 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 26478	. . . . 5 ⊢ (𝑎 ∈ 𝔸 → 𝑎 ∈ ℂ)
7		mpaacl 43900	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → (minPolyAA‘𝑎) ∈ (Poly‘ℚ))
8		mpaaroot 43902	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → ((minPolyAA‘𝑎)‘𝑎) = 0)
9		mpaadgr 43901	. . . . . . . 8 ⊢ (𝑎 ∈ 𝔸 → (deg‘(minPolyAA‘𝑎)) = (degAA‘𝑎))
10	9	fveq2d 6885	. . . . . . 7 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = ((coeff‘(minPolyAA‘𝑎))‘(degAA‘𝑎)))
11		mpaamn 43903	. . . . . . 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 6880	. . . . . . . . 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 6881	. . . . . . . . . 10 ⊢ (𝑏 = (minPolyAA‘𝑎) → (coeff‘𝑏) = (coeff‘(minPolyAA‘𝑎)))
16		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑏 = (minPolyAA‘𝑎) → (deg‘𝑏) = (deg‘(minPolyAA‘𝑎)))
17	15, 16	fveq12d 6888	. . . . . . . . 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 643	. . . . . . 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 3581	. . . . . 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 849	. . . . 5 ⊢ (𝑎 ∈ 𝔸 → ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1))
22	6, 21	jca 520	. . . 4 ⊢ (𝑎 ∈ 𝔸 → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
23		simpl 487	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ (Poly‘ℚ))
24		coe0 26413	. . . . . . . . . . . . . . 15 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
25	24	fveq1i 6882	. . . . . . . . . . . . . 14 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) = ((ℕ0 × {0})‘(deg‘0𝑝))
26		dgr0 26419	. . . . . . . . . . . . . . . 16 ⊢ (deg‘0𝑝) = 0
27		0nn0 12514	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ0
28	26, 27	eqeltri 2859	. . . . . . . . . . . . . . 15 ⊢ (deg‘0𝑝) ∈ ℕ0
29		c0ex 11195	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
30	29	fvconst2 7202	. . . . . . . . . . . . . . 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 12307	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
34	32, 33	eqnetri 3028	. . . . . . . . . . . 12 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1
35		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 0𝑝 → (coeff‘𝑏) = (coeff‘0𝑝))
36		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 0𝑝 → (deg‘𝑏) = (deg‘0𝑝))
37	35, 36	fveq12d 6888	. . . . . . . . . . . . 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 261	. . . . . . . . . . 11 ⊢ (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1)
40	39	necon2i 2992	. . . . . . . . . 10 ⊢ (((coeff‘𝑏)‘(deg‘𝑏)) = 1 → 𝑏 ≠ 0𝑝)
41	40	ad2antll 741	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ≠ 0𝑝)
42		eldifsn 4753	. . . . . . . . 9 ⊢ (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ (𝑏 ∈ (Poly‘ℚ) ∧ 𝑏 ≠ 0𝑝))
43	23, 41, 42	sylanbrc 594	. . . . . . . 8 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}))
44		simprl 782	. . . . . . . 8 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏‘𝑎) = 0)
45	43, 44	jca 520	. . . . . . 7 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ (𝑏‘𝑎) = 0))
46	45	reximi2 3098	. . . . . 6 ⊢ (∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏‘𝑎) = 0)
47	46	anim2i 628	. . . . 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 26483	. . . . 5 ⊢ (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏‘𝑎) = 0))
49	47, 48	sylibr 237	. . . 4 ⊢ ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑎 ∈ 𝔸)
50	22, 49	impbii 212	. . 3 ⊢ (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
51	1, 5, 50	3bitr4ri 307	. 2 ⊢ (𝑎 ∈ 𝔸 ↔ 𝑎 ∈ (IntgOver‘ℚ))
52	51	eqriv 2760	1 ⊢ 𝔸 = (IntgOver‘ℚ)

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  {crab 3416   ∖ cdif 3902   ⊆ wss 3905  {csn 4589   × cxp 5659  ‘cfv 6536  ℂcc 11093  0cc0 11095  1c1 11096  ℕ₀cn0 12499  ℚcq 12967  0_𝑝c0p 25828  Polycply 26341  coeffccoe 26343  degcdgr 26344  𝔸caa 26475  deg_AAcdgraa 43887  minPolyAAcmpaa 43888  IntgOvercitgo 43904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-q 12968  df-rp 13012  df-fz 13531  df-fzo 13679  df-fl 13821  df-mod 13899  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-rlim 15536  df-sum 15734  df-0p 25829  df-ply 26345  df-coe 26347  df-dgr 26348  df-aa 26476  df-dgraa 43889  df-mpaa 43890  df-itgo 43906

This theorem is referenced by: (None)