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Mirrors > Home > MPE Home > Th. List > Mathboxes > itgocn | Structured version Visualization version GIF version |
Description: All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
itgocn | ⊢ (IntgOver‘𝑆) ⊆ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-itgo 36748 | . . . . 5 ⊢ IntgOver = (𝑎 ∈ 𝒫 ℂ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ (Poly‘𝑎)((𝑐‘𝑏) = 0 ∧ ((coeff‘𝑐)‘(deg‘𝑐)) = 1)}) | |
2 | 1 | dmmptss 5548 | . . . 4 ⊢ dom IntgOver ⊆ 𝒫 ℂ |
3 | 2 | sseli 3564 | . . 3 ⊢ (𝑆 ∈ dom IntgOver → 𝑆 ∈ 𝒫 ℂ) |
4 | cnex 9896 | . . . . 5 ⊢ ℂ ∈ V | |
5 | 4 | elpw2 4755 | . . . 4 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ) |
6 | itgoval 36750 | . . . . 5 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)}) | |
7 | ssrab2 3650 | . . . . 5 ⊢ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ ℂ | |
8 | 6, 7 | syl6eqss 3618 | . . . 4 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) ⊆ ℂ) |
9 | 5, 8 | sylbi 206 | . . 3 ⊢ (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) ⊆ ℂ) |
10 | 3, 9 | syl 17 | . 2 ⊢ (𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ) |
11 | ndmfv 6128 | . . 3 ⊢ (¬ 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) = ∅) | |
12 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ ℂ | |
13 | 11, 12 | syl6eqss 3618 | . 2 ⊢ (¬ 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ) |
14 | 10, 13 | pm2.61i 175 | 1 ⊢ (IntgOver‘𝑆) ⊆ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 dom cdm 5038 ‘cfv 5804 ℂcc 9813 0cc0 9815 1c1 9816 Polycply 23744 coeffccoe 23746 degcdgr 23747 IntgOvercitgo 36746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-cnex 9871 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-itgo 36748 |
This theorem is referenced by: (None) |
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