Theorem itgocn 39771

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 39766	. . . . 5 ⊢ IntgOver = (𝑎 ∈ 𝒫 ℂ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ (Poly‘𝑎)((𝑐‘𝑏) = 0 ∧ ((coeff‘𝑐)‘(deg‘𝑐)) = 1)})
2	1	dmmptss 6097	. . . 4 ⊢ dom IntgOver ⊆ 𝒫 ℂ
3	2	sseli 3965	. . 3 ⊢ (𝑆 ∈ dom IntgOver → 𝑆 ∈ 𝒫 ℂ)
4		cnex 10620	. . . . 5 ⊢ ℂ ∈ V
5	4	elpw2 5250	. . . 4 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
6		itgoval 39768	. . . . 5 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
7		ssrab2 4058	. . . . 5 ⊢ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ ℂ
8	6, 7	eqsstrdi 4023	. . . 4 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) ⊆ ℂ)
9	5, 8	sylbi 219	. . 3 ⊢ (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) ⊆ ℂ)
10	3, 9	syl 17	. 2 ⊢ (𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
11		ndmfv 6702	. . 3 ⊢ (¬ 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) = ∅)
12		0ss 4352	. . 3 ⊢ ∅ ⊆ ℂ
13	11, 12	eqsstrdi 4023	. 2 ⊢ (¬ 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
14	10, 13	pm2.61i 184	1 ⊢ (IntgOver‘𝑆) ⊆ ℂ

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  {crab 3144   ⊆ wss 3938  ∅c0 4293  𝒫 cpw 4541  dom cdm 5557  ‘cfv 6357  ℂcc 10537  0cc0 10539  1c1 10540  Polycply 24776  coeffccoe 24778  degcdgr 24779  IntgOvercitgo 39764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-cnex 10595

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fv 6365  df-itgo 39766

This theorem is referenced by: (None)