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Mirrors > Home > MPE Home > Th. List > rennim | Structured version Visualization version GIF version |
Description: A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.) |
Ref | Expression |
---|---|
rennim | ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∉ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 11219 | . . . . . . 7 ⊢ i ∈ ℂ | |
2 | recn 11250 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | mulcl 11244 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
4 | 1, 2, 3 | sylancr 585 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
5 | rpre 13038 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℝ+ → (i · 𝐴) ∈ ℝ) | |
6 | rereb 15127 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℂ → ((i · 𝐴) ∈ ℝ ↔ (ℜ‘(i · 𝐴)) = (i · 𝐴))) | |
7 | 5, 6 | imbitrid 243 | . . . . . 6 ⊢ ((i · 𝐴) ∈ ℂ → ((i · 𝐴) ∈ ℝ+ → (ℜ‘(i · 𝐴)) = (i · 𝐴))) |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((i · 𝐴) ∈ ℝ+ → (ℜ‘(i · 𝐴)) = (i · 𝐴))) |
9 | 4 | addlidd 11467 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (0 + (i · 𝐴)) = (i · 𝐴)) |
10 | 9 | fveq2d 6907 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (ℜ‘(0 + (i · 𝐴))) = (ℜ‘(i · 𝐴))) |
11 | 0re 11268 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
12 | crre 15121 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (ℜ‘(0 + (i · 𝐴))) = 0) | |
13 | 11, 12 | mpan 688 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (ℜ‘(0 + (i · 𝐴))) = 0) |
14 | 10, 13 | eqtr3d 2768 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (ℜ‘(i · 𝐴)) = 0) |
15 | 14 | eqeq1d 2728 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((ℜ‘(i · 𝐴)) = (i · 𝐴) ↔ 0 = (i · 𝐴))) |
16 | 8, 15 | sylibd 238 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((i · 𝐴) ∈ ℝ+ → 0 = (i · 𝐴))) |
17 | rpne0 13046 | . . . . . 6 ⊢ ((i · 𝐴) ∈ ℝ+ → (i · 𝐴) ≠ 0) | |
18 | 17 | necon2bi 2961 | . . . . 5 ⊢ ((i · 𝐴) = 0 → ¬ (i · 𝐴) ∈ ℝ+) |
19 | 18 | eqcoms 2734 | . . . 4 ⊢ (0 = (i · 𝐴) → ¬ (i · 𝐴) ∈ ℝ+) |
20 | 16, 19 | syl6 35 | . . 3 ⊢ (𝐴 ∈ ℝ → ((i · 𝐴) ∈ ℝ+ → ¬ (i · 𝐴) ∈ ℝ+)) |
21 | 20 | pm2.01d 189 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ (i · 𝐴) ∈ ℝ+) |
22 | df-nel 3037 | . 2 ⊢ ((i · 𝐴) ∉ ℝ+ ↔ ¬ (i · 𝐴) ∈ ℝ+) | |
23 | 21, 22 | sylibr 233 | 1 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∉ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 ∉ wnel 3036 ‘cfv 6556 (class class class)co 7426 ℂcc 11158 ℝcr 11159 0cc0 11160 ici 11162 + caddc 11163 · cmul 11165 ℝ+crp 13030 ℜcre 15104 |
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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-po 5596 df-so 5597 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-2 12329 df-rp 13031 df-cj 15106 df-re 15107 df-im 15108 |
This theorem is referenced by: sqrt0 15248 resqreu 15259 resqrtcl 15260 |
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