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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 10596 | . . . . . . 7 ⊢ i ∈ ℂ | |
2 | recn 10627 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | mulcl 10621 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
4 | 1, 2, 3 | sylancr 589 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
5 | rpre 12398 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℝ+ → (i · 𝐴) ∈ ℝ) | |
6 | rereb 14479 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℂ → ((i · 𝐴) ∈ ℝ ↔ (ℜ‘(i · 𝐴)) = (i · 𝐴))) | |
7 | 5, 6 | syl5ib 246 | . . . . . 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 10841 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (0 + (i · 𝐴)) = (i · 𝐴)) |
10 | 9 | fveq2d 6674 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (ℜ‘(0 + (i · 𝐴))) = (ℜ‘(i · 𝐴))) |
11 | 0re 10643 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
12 | crre 14473 | . . . . . . . 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 2858 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (ℜ‘(i · 𝐴)) = 0) |
15 | 14 | eqeq1d 2823 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((ℜ‘(i · 𝐴)) = (i · 𝐴) ↔ 0 = (i · 𝐴))) |
16 | 8, 15 | sylibd 241 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((i · 𝐴) ∈ ℝ+ → 0 = (i · 𝐴))) |
17 | rpne0 12406 | . . . . . 6 ⊢ ((i · 𝐴) ∈ ℝ+ → (i · 𝐴) ≠ 0) | |
18 | 17 | necon2bi 3046 | . . . . 5 ⊢ ((i · 𝐴) = 0 → ¬ (i · 𝐴) ∈ ℝ+) |
19 | 18 | eqcoms 2829 | . . . 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 192 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ (i · 𝐴) ∈ ℝ+) |
22 | df-nel 3124 | . 2 ⊢ ((i · 𝐴) ∉ ℝ+ ↔ ¬ (i · 𝐴) ∈ ℝ+) | |
23 | 21, 22 | sylibr 236 | 1 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∉ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 ici 10539 + caddc 10540 · cmul 10542 ℝ+crp 12390 ℜcre 14456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-rp 12391 df-cj 14458 df-re 14459 df-im 14460 |
This theorem is referenced by: sqr0lem 14600 resqreu 14612 resqrtcl 14613 |
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