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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 41389 | . . . . 5 ⊢ IntgOver = (𝑎 ∈ 𝒫 ℂ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ (Poly‘𝑎)((𝑐‘𝑏) = 0 ∧ ((coeff‘𝑐)‘(deg‘𝑐)) = 1)}) | |
2 | 1 | dmmptss 6192 | . . . 4 ⊢ dom IntgOver ⊆ 𝒫 ℂ |
3 | 2 | sseli 3939 | . . 3 ⊢ (𝑆 ∈ dom IntgOver → 𝑆 ∈ 𝒫 ℂ) |
4 | cnex 11091 | . . . . 5 ⊢ ℂ ∈ V | |
5 | 4 | elpw2 5301 | . . . 4 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ) |
6 | itgoval 41391 | . . . . 5 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)}) | |
7 | ssrab2 4036 | . . . . 5 ⊢ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ ℂ | |
8 | 6, 7 | eqsstrdi 3997 | . . . 4 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) ⊆ ℂ) |
9 | 5, 8 | sylbi 216 | . . 3 ⊢ (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) ⊆ ℂ) |
10 | 3, 9 | syl 17 | . 2 ⊢ (𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ) |
11 | ndmfv 6875 | . . 3 ⊢ (¬ 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) = ∅) | |
12 | 0ss 4355 | . . 3 ⊢ ∅ ⊆ ℂ | |
13 | 11, 12 | eqsstrdi 3997 | . 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 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3072 {crab 3406 ⊆ wss 3909 ∅c0 4281 𝒫 cpw 4559 dom cdm 5632 ‘cfv 6494 ℂcc 11008 0cc0 11010 1c1 11011 Polycply 25497 coeffccoe 25499 degcdgr 25500 IntgOvercitgo 41387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-cnex 11066 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fv 6502 df-itgo 41389 |
This theorem is referenced by: (None) |
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