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.

Step	Hyp	Ref	Expression
1		rabid 3312	. . 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 12173	. . . . 5 ⊢ ℚ ⊆ ℂ
3		itgoval 39191	. . . . 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 2852	. . 3 ⊢ (𝑎 ∈ (IntgOver‘ℚ) ↔ 𝑎 ∈ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 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)) = (degAA‘𝑎))
10	9	fveq2d 6501	. . . . . . 7 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = ((coeff‘(minPolyAA‘𝑎))‘(degAA‘𝑎)))
11		mpaamn 39186	. . . . . . 7 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(degAA‘𝑎)) = 1)
12	10, 11	eqtrd 2809	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)
13		fveq1 6496	. . . . . . . . 9 ⊢ (𝑏 = (minPolyAA‘𝑎) → (𝑏‘𝑎) = ((minPolyAA‘𝑎)‘𝑎))
14	13	eqeq1d 2775	. . . . . . . 8 ⊢ (𝑏 = (minPolyAA‘𝑎) → ((𝑏‘𝑎) = 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)))
17	15, 16	fveq12d 6504	. . . . . . . . 9 ⊢ (𝑏 = (minPolyAA‘𝑎) → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))))
18	17	eqeq1d 2775	. . . . . . . 8 ⊢ (𝑏 = (minPolyAA‘𝑎) → (((coeff‘𝑏)‘(deg‘𝑏)) = 1 ↔ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1))
19	14, 18	anbi12d 622	. . . . . . 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 3530	. . . . . 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 825	. . . . 5 ⊢ (𝑎 ∈ 𝔸 → ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1))
22	6, 21	jca 504	. . . 4 ⊢ (𝑎 ∈ 𝔸 → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
23		simpl 475	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ (Poly‘ℚ))
24		coe0 24565	. . . . . . . . . . . . . . 15 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
25	24	fveq1i 6498	. . . . . . . . . . . . . 14 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) = ((ℕ0 × {0})‘(deg‘0𝑝))
26		dgr0 24571	. . . . . . . . . . . . . . . 16 ⊢ (deg‘0𝑝) = 0
27		0nn0 11723	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ0
28	26, 27	eqeltri 2857	. . . . . . . . . . . . . . 15 ⊢ (deg‘0𝑝) ∈ ℕ0
29		c0ex 10432	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
30	29	fvconst2 6792	. . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . 13 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) = 0
33		0ne1 11510	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
34	32, 33	eqnetri 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))
37	35, 36	fveq12d 6504	. . . . . . . . . . . . 13 ⊢ (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘0𝑝)‘(deg‘0𝑝)))
38	37	neeq1d 3021	. . . . . . . . . . . 12 ⊢ (𝑏 = 0𝑝 → (((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1 ↔ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1))
39	34, 38	mpbiri 250	. . . . . . . . . . 11 ⊢ (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1)
40	39	necon2i 2996	. . . . . . . . . 10 ⊢ (((coeff‘𝑏)‘(deg‘𝑏)) = 1 → 𝑏 ≠ 0𝑝)
41	40	ad2antll 717	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ≠ 0𝑝)
42		eldifsn 4590	. . . . . . . . 9 ⊢ (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ (𝑏 ∈ (Poly‘ℚ) ∧ 𝑏 ≠ 0𝑝))
43	23, 41, 42	sylanbrc 575	. . . . . . . 8 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}))
44		simprl 759	. . . . . . . 8 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏‘𝑎) = 0)
45	43, 44	jca 504	. . . . . . 7 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ (𝑏‘𝑎) = 0))
46	45	reximi2 3186	. . . . . 6 ⊢ (∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏‘𝑎) = 0)
47	46	anim2i 608	. . . . 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 24630	. . . . 5 ⊢ (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏‘𝑎) = 0))
49	47, 48	sylibr 226	. . . 4 ⊢ ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑎 ∈ 𝔸)
50	22, 49	impbii 201	. . 3 ⊢ (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
51	1, 5, 50	3bitr4ri 296	. 2 ⊢ (𝑎 ∈ 𝔸 ↔ 𝑎 ∈ (IntgOver‘ℚ))
52	51	eqriv 2770	1 ⊢ 𝔸 = (IntgOver‘ℚ)

Syntax hints:   ∧ wa 387   = wceq 1508   ∈ wcel 2051   ≠ wne 2962  ∃wrex 3084  {crab 3087   ∖ cdif 3821   ⊆ wss 3824  {csn 4436   × cxp 5402  ‘cfv 6186  ℂcc 10332  0cc0 10334  1c1 10335  ℕ₀cn0 11706  ℚcq 12161  0_𝑝c0p 23989  Polycply 24493  coeffccoe 24495  degcdgr 24496  𝔸caa 24622  deg_AAcdgraa 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)