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Mathbox for Stefan O'Rear |
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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 43102 | . . . . 5 ⊢ IntgOver = (𝑎 ∈ 𝒫 ℂ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ (Poly‘𝑎)((𝑐‘𝑏) = 0 ∧ ((coeff‘𝑐)‘(deg‘𝑐)) = 1)}) | |
2 | 1 | dmmptss 6257 | . . . 4 ⊢ dom IntgOver ⊆ 𝒫 ℂ |
3 | 2 | sseli 3991 | . . 3 ⊢ (𝑆 ∈ dom IntgOver → 𝑆 ∈ 𝒫 ℂ) |
4 | cnex 11227 | . . . . 5 ⊢ ℂ ∈ V | |
5 | 4 | elpw2 5335 | . . . 4 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ) |
6 | itgoval 43104 | . . . . 5 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)}) | |
7 | ssrab2 4090 | . . . . 5 ⊢ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ ℂ | |
8 | 6, 7 | eqsstrdi 4050 | . . . 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 6936 | . . 3 ⊢ (¬ 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) = ∅) | |
12 | 0ss 4405 | . . 3 ⊢ ∅ ⊆ ℂ | |
13 | 11, 12 | eqsstrdi 4050 | . 2 ⊢ (¬ 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ) |
14 | 10, 13 | pm2.61i 182 | 1 ⊢ (IntgOver‘𝑆) ⊆ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1535 ∈ wcel 2104 ∃wrex 3066 {crab 3432 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4604 dom cdm 5683 ‘cfv 6558 ℂcc 11144 0cc0 11146 1c1 11147 Polycply 26219 coeffccoe 26221 degcdgr 26222 IntgOvercitgo 43100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 ax-cnex 11202 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 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 6510 df-fun 6560 df-fv 6566 df-itgo 43102 |
This theorem is referenced by: (None) |
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