Theorem itgocn 43154

Description: All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Assertion

Ref	Expression
itgocn	⊢ (IntgOver‘𝑆) ⊆ ℂ

Proof of Theorem itgocn

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-itgo 43149	. . . . 5 ⊢ IntgOver = (𝑎 ∈ 𝒫 ℂ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ (Poly‘𝑎)((𝑐‘𝑏) = 0 ∧ ((coeff‘𝑐)‘(deg‘𝑐)) = 1)})
2	1	dmmptss 6259	. . . 4 ⊢ dom IntgOver ⊆ 𝒫 ℂ
3	2	sseli 3978	. . 3 ⊢ (𝑆 ∈ dom IntgOver → 𝑆 ∈ 𝒫 ℂ)
4		cnex 11232	. . . . 5 ⊢ ℂ ∈ V
5	4	elpw2 5332	. . . 4 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
6		itgoval 43151	. . . . 5 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
7		ssrab2 4079	. . . . 5 ⊢ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ ℂ
8	6, 7	eqsstrdi 4027	. . . 4 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) ⊆ ℂ)
9	5, 8	sylbi 217	. . 3 ⊢ (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) ⊆ ℂ)
10	3, 9	syl 17	. 2 ⊢ (𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
11		ndmfv 6939	. . 3 ⊢ (¬ 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) = ∅)
12		0ss 4399	. . 3 ⊢ ∅ ⊆ ℂ
13	11, 12	eqsstrdi 4027	. 2 ⊢ (¬ 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
14	10, 13	pm2.61i 182	1 ⊢ (IntgOver‘𝑆) ⊆ ℂ

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3069  {crab 3435   ⊆ wss 3950  ∅c0 4332  𝒫 cpw 4598  dom cdm 5683  ‘cfv 6559  ℂcc 11149  0cc0 11151  1c1 11152  Polycply 26213  coeffccoe 26215  degcdgr 26216  IntgOvercitgo 43147

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430  ax-cnex 11207

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fv 6567  df-itgo 43149

This theorem is referenced by: (None)