Theorem aaitgo 43616

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 3412	. . 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 12902	. . . . 5 ⊢ ℚ ⊆ ℂ
3		itgoval 43615	. . . . 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 2831	. . 3 ⊢ (𝑎 ∈ (IntgOver‘ℚ) ↔ 𝑎 ∈ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
6		aacn 26302	. . . . 5 ⊢ (𝑎 ∈ 𝔸 → 𝑎 ∈ ℂ)
7		mpaacl 43607	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → (minPolyAA‘𝑎) ∈ (Poly‘ℚ))
8		mpaaroot 43609	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → ((minPolyAA‘𝑎)‘𝑎) = 0)
9		mpaadgr 43608	. . . . . . . 8 ⊢ (𝑎 ∈ 𝔸 → (deg‘(minPolyAA‘𝑎)) = (degAA‘𝑎))
10	9	fveq2d 6832	. . . . . . 7 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = ((coeff‘(minPolyAA‘𝑎))‘(degAA‘𝑎)))
11		mpaamn 43610	. . . . . . 7 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(degAA‘𝑎)) = 1)
12	10, 11	eqtrd 2774	. . . . . 6 ⊢ (𝑎 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1)
13		fveq1 6827	. . . . . . . . 9 ⊢ (𝑏 = (minPolyAA‘𝑎) → (𝑏‘𝑎) = ((minPolyAA‘𝑎)‘𝑎))
14	13	eqeq1d 2741	. . . . . . . 8 ⊢ (𝑏 = (minPolyAA‘𝑎) → ((𝑏‘𝑎) = 0 ↔ ((minPolyAA‘𝑎)‘𝑎) = 0))
15		fveq2 6828	. . . . . . . . . 10 ⊢ (𝑏 = (minPolyAA‘𝑎) → (coeff‘𝑏) = (coeff‘(minPolyAA‘𝑎)))
16		fveq2 6828	. . . . . . . . . 10 ⊢ (𝑏 = (minPolyAA‘𝑎) → (deg‘𝑏) = (deg‘(minPolyAA‘𝑎)))
17	15, 16	fveq12d 6835	. . . . . . . . 9 ⊢ (𝑏 = (minPolyAA‘𝑎) → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))))
18	17	eqeq1d 2741	. . . . . . . 8 ⊢ (𝑏 = (minPolyAA‘𝑎) → (((coeff‘𝑏)‘(deg‘𝑏)) = 1 ↔ ((coeff‘(minPolyAA‘𝑎))‘(deg‘(minPolyAA‘𝑎))) = 1))
19	14, 18	anbi12d 638	. . . . . . 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 3560	. . . . . 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 842	. . . . 5 ⊢ (𝑎 ∈ 𝔸 → ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1))
22	6, 21	jca 516	. . . 4 ⊢ (𝑎 ∈ 𝔸 → (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
23		simpl 483	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ (Poly‘ℚ))
24		coe0 26240	. . . . . . . . . . . . . . 15 ⊢ (coeff‘0𝑝) = (ℕ0 × {0})
25	24	fveq1i 6829	. . . . . . . . . . . . . 14 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) = ((ℕ0 × {0})‘(deg‘0𝑝))
26		dgr0 26246	. . . . . . . . . . . . . . . 16 ⊢ (deg‘0𝑝) = 0
27		0nn0 12444	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ0
28	26, 27	eqeltri 2835	. . . . . . . . . . . . . . 15 ⊢ (deg‘0𝑝) ∈ ℕ0
29		c0ex 11130	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
30	29	fvconst2 7149	. . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . 13 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) = 0
33		0ne1 12244	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
34	32, 33	eqnetri 3004	. . . . . . . . . . . 12 ⊢ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1
35		fveq2 6828	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 0𝑝 → (coeff‘𝑏) = (coeff‘0𝑝))
36		fveq2 6828	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 0𝑝 → (deg‘𝑏) = (deg‘0𝑝))
37	35, 36	fveq12d 6835	. . . . . . . . . . . . 13 ⊢ (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) = ((coeff‘0𝑝)‘(deg‘0𝑝)))
38	37	neeq1d 2993	. . . . . . . . . . . 12 ⊢ (𝑏 = 0𝑝 → (((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1 ↔ ((coeff‘0𝑝)‘(deg‘0𝑝)) ≠ 1))
39	34, 38	mpbiri 259	. . . . . . . . . . 11 ⊢ (𝑏 = 0𝑝 → ((coeff‘𝑏)‘(deg‘𝑏)) ≠ 1)
40	39	necon2i 2968	. . . . . . . . . 10 ⊢ (((coeff‘𝑏)‘(deg‘𝑏)) = 1 → 𝑏 ≠ 0𝑝)
41	40	ad2antll 735	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ≠ 0𝑝)
42		eldifsn 4720	. . . . . . . . 9 ⊢ (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ (𝑏 ∈ (Poly‘ℚ) ∧ 𝑏 ≠ 0𝑝))
43	23, 41, 42	sylanbrc 589	. . . . . . . 8 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}))
44		simprl 776	. . . . . . . 8 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏‘𝑎) = 0)
45	43, 44	jca 516	. . . . . . 7 ⊢ ((𝑏 ∈ (Poly‘ℚ) ∧ ((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → (𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ (𝑏‘𝑎) = 0))
46	45	reximi2 3072	. . . . . 6 ⊢ (∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1) → ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏‘𝑎) = 0)
47	46	anim2i 623	. . . . 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 26307	. . . . 5 ⊢ (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑏‘𝑎) = 0))
49	47, 48	sylibr 235	. . . 4 ⊢ ((𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)) → 𝑎 ∈ 𝔸)
50	22, 49	impbii 210	. . 3 ⊢ (𝑎 ∈ 𝔸 ↔ (𝑎 ∈ ℂ ∧ ∃𝑏 ∈ (Poly‘ℚ)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)))
51	1, 5, 50	3bitr4ri 305	. 2 ⊢ (𝑎 ∈ 𝔸 ↔ 𝑎 ∈ (IntgOver‘ℚ))
52	51	eqriv 2736	1 ⊢ 𝔸 = (IntgOver‘ℚ)

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063  {crab 3391   ∖ cdif 3880   ⊆ wss 3883  {csn 4556   × cxp 5617  ‘cfv 6486  ℂcc 11028  0cc0 11030  1c1 11031  ℕ₀cn0 12429  ℚcq 12890  0_𝑝c0p 25655  Polycply 26168  coeffccoe 26170  degcdgr 26171  𝔸caa 26299  deg_AAcdgraa 43594  minPolyAAcmpaa 43595  IntgOvercitgo 43611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7621  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-pm 8767  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9346  df-inf 9347  df-oi 9416  df-card 9855  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-div 11800  df-nn 12167  df-2 12236  df-3 12237  df-n0 12430  df-z 12517  df-uz 12781  df-q 12891  df-rp 12935  df-fz 13454  df-fzo 13601  df-fl 13743  df-mod 13821  df-seq 13956  df-exp 14016  df-hash 14285  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15442  df-rlim 15443  df-sum 15641  df-0p 25656  df-ply 26172  df-coe 26174  df-dgr 26175  df-aa 26300  df-dgraa 43596  df-mpaa 43597  df-itgo 43613

This theorem is referenced by: (None)