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Mirrors > Home > ILE Home > Th. List > rennim | 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 7708 | . . . . . . 7 ⊢ i ∈ ℂ | |
2 | recn 7746 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | mulcl 7740 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
4 | 1, 2, 3 | sylancr 410 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
5 | rpre 9441 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℝ+ → (i · 𝐴) ∈ ℝ) | |
6 | rereb 10628 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℂ → ((i · 𝐴) ∈ ℝ ↔ (ℜ‘(i · 𝐴)) = (i · 𝐴))) | |
7 | 5, 6 | syl5ib 153 | . . . . . 6 ⊢ ((i · 𝐴) ∈ ℂ → ((i · 𝐴) ∈ ℝ+ → (ℜ‘(i · 𝐴)) = (i · 𝐴))) |
8 | 4, 7 | syl 14 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((i · 𝐴) ∈ ℝ+ → (ℜ‘(i · 𝐴)) = (i · 𝐴))) |
9 | 4 | addid2d 7905 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (0 + (i · 𝐴)) = (i · 𝐴)) |
10 | 9 | fveq2d 5418 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (ℜ‘(0 + (i · 𝐴))) = (ℜ‘(i · 𝐴))) |
11 | 0re 7759 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
12 | crre 10622 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (ℜ‘(0 + (i · 𝐴))) = 0) | |
13 | 11, 12 | mpan 420 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (ℜ‘(0 + (i · 𝐴))) = 0) |
14 | 10, 13 | eqtr3d 2172 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (ℜ‘(i · 𝐴)) = 0) |
15 | 14 | eqeq1d 2146 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((ℜ‘(i · 𝐴)) = (i · 𝐴) ↔ 0 = (i · 𝐴))) |
16 | 8, 15 | sylibd 148 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((i · 𝐴) ∈ ℝ+ → 0 = (i · 𝐴))) |
17 | rpne0 9450 | . . . . . 6 ⊢ ((i · 𝐴) ∈ ℝ+ → (i · 𝐴) ≠ 0) | |
18 | 17 | necon2bi 2361 | . . . . 5 ⊢ ((i · 𝐴) = 0 → ¬ (i · 𝐴) ∈ ℝ+) |
19 | 18 | eqcoms 2140 | . . . 4 ⊢ (0 = (i · 𝐴) → ¬ (i · 𝐴) ∈ ℝ+) |
20 | 16, 19 | syl6 33 | . . 3 ⊢ (𝐴 ∈ ℝ → ((i · 𝐴) ∈ ℝ+ → ¬ (i · 𝐴) ∈ ℝ+)) |
21 | 20 | pm2.01d 607 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ (i · 𝐴) ∈ ℝ+) |
22 | df-nel 2402 | . 2 ⊢ ((i · 𝐴) ∉ ℝ+ ↔ ¬ (i · 𝐴) ∈ ℝ+) | |
23 | 21, 22 | sylibr 133 | 1 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∉ ℝ+) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ∈ wcel 1480 ∉ wnel 2401 ‘cfv 5118 (class class class)co 5767 ℂcc 7611 ℝcr 7612 0cc0 7613 ici 7615 + caddc 7616 · cmul 7618 ℝ+crp 9434 ℜcre 10605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-2 8772 df-rp 9435 df-cj 10607 df-re 10608 df-im 10609 |
This theorem is referenced by: (None) |
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