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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 39100 | . . . . 5 ⊢ IntgOver = (𝑎 ∈ 𝒫 ℂ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ (Poly‘𝑎)((𝑐‘𝑏) = 0 ∧ ((coeff‘𝑐)‘(deg‘𝑐)) = 1)}) | |
2 | 1 | dmmptss 5928 | . . . 4 ⊢ dom IntgOver ⊆ 𝒫 ℂ |
3 | 2 | sseli 3850 | . . 3 ⊢ (𝑆 ∈ dom IntgOver → 𝑆 ∈ 𝒫 ℂ) |
4 | cnex 10408 | . . . . 5 ⊢ ℂ ∈ V | |
5 | 4 | elpw2 5098 | . . . 4 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ) |
6 | itgoval 39102 | . . . . 5 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)}) | |
7 | ssrab2 3942 | . . . . 5 ⊢ {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏‘𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ ℂ | |
8 | 6, 7 | syl6eqss 3907 | . . . 4 ⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) ⊆ ℂ) |
9 | 5, 8 | sylbi 209 | . . 3 ⊢ (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) ⊆ ℂ) |
10 | 3, 9 | syl 17 | . 2 ⊢ (𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ) |
11 | ndmfv 6523 | . . 3 ⊢ (¬ 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) = ∅) | |
12 | 0ss 4230 | . . 3 ⊢ ∅ ⊆ ℂ | |
13 | 11, 12 | syl6eqss 3907 | . 2 ⊢ (¬ 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ) |
14 | 10, 13 | pm2.61i 177 | 1 ⊢ (IntgOver‘𝑆) ⊆ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∃wrex 3083 {crab 3086 ⊆ wss 3825 ∅c0 4173 𝒫 cpw 4416 dom cdm 5400 ‘cfv 6182 ℂcc 10325 0cc0 10327 1c1 10328 Polycply 24467 coeffccoe 24469 degcdgr 24470 IntgOvercitgo 39098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-cnex 10383 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fv 6190 df-itgo 39100 |
This theorem is referenced by: (None) |
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