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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 10201 | . . . . . . 7 ⊢ i ∈ ℂ | |
2 | recn 10232 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | mulcl 10226 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
4 | 1, 2, 3 | sylancr 575 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
5 | rpre 12042 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℝ+ → (i · 𝐴) ∈ ℝ) | |
6 | rereb 14068 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℂ → ((i · 𝐴) ∈ ℝ ↔ (ℜ‘(i · 𝐴)) = (i · 𝐴))) | |
7 | 5, 6 | syl5ib 234 | . . . . . 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 | addid2d 10443 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (0 + (i · 𝐴)) = (i · 𝐴)) |
10 | 9 | fveq2d 6337 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (ℜ‘(0 + (i · 𝐴))) = (ℜ‘(i · 𝐴))) |
11 | 0re 10246 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
12 | crre 14062 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (ℜ‘(0 + (i · 𝐴))) = 0) | |
13 | 11, 12 | mpan 670 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (ℜ‘(0 + (i · 𝐴))) = 0) |
14 | 10, 13 | eqtr3d 2807 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (ℜ‘(i · 𝐴)) = 0) |
15 | 14 | eqeq1d 2773 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((ℜ‘(i · 𝐴)) = (i · 𝐴) ↔ 0 = (i · 𝐴))) |
16 | 8, 15 | sylibd 229 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((i · 𝐴) ∈ ℝ+ → 0 = (i · 𝐴))) |
17 | rpne0 12051 | . . . . . 6 ⊢ ((i · 𝐴) ∈ ℝ+ → (i · 𝐴) ≠ 0) | |
18 | 17 | necon2bi 2973 | . . . . 5 ⊢ ((i · 𝐴) = 0 → ¬ (i · 𝐴) ∈ ℝ+) |
19 | 18 | eqcoms 2779 | . . . 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 181 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ (i · 𝐴) ∈ ℝ+) |
22 | df-nel 3047 | . 2 ⊢ ((i · 𝐴) ∉ ℝ+ ↔ ¬ (i · 𝐴) ∈ ℝ+) | |
23 | 21, 22 | sylibr 224 | 1 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∉ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1631 ∈ wcel 2145 ∉ wnel 3046 ‘cfv 6030 (class class class)co 6796 ℂcc 10140 ℝcr 10141 0cc0 10142 ici 10144 + caddc 10145 · cmul 10147 ℝ+crp 12035 ℜcre 14045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-po 5171 df-so 5172 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-2 11285 df-rp 12036 df-cj 14047 df-re 14048 df-im 14049 |
This theorem is referenced by: sqr0lem 14189 resqreu 14201 resqrtcl 14202 |
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