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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 10393 | . . . . . . 7 ⊢ i ∈ ℂ | |
2 | recn 10424 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | mulcl 10418 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
4 | 1, 2, 3 | sylancr 579 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∈ ℂ) |
5 | rpre 12211 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℝ+ → (i · 𝐴) ∈ ℝ) | |
6 | rereb 14339 | . . . . . . 7 ⊢ ((i · 𝐴) ∈ ℂ → ((i · 𝐴) ∈ ℝ ↔ (ℜ‘(i · 𝐴)) = (i · 𝐴))) | |
7 | 5, 6 | syl5ib 236 | . . . . . 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 10640 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (0 + (i · 𝐴)) = (i · 𝐴)) |
10 | 9 | fveq2d 6501 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (ℜ‘(0 + (i · 𝐴))) = (ℜ‘(i · 𝐴))) |
11 | 0re 10440 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
12 | crre 14333 | . . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (ℜ‘(0 + (i · 𝐴))) = 0) | |
13 | 11, 12 | mpan 678 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (ℜ‘(0 + (i · 𝐴))) = 0) |
14 | 10, 13 | eqtr3d 2811 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (ℜ‘(i · 𝐴)) = 0) |
15 | 14 | eqeq1d 2775 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((ℜ‘(i · 𝐴)) = (i · 𝐴) ↔ 0 = (i · 𝐴))) |
16 | 8, 15 | sylibd 231 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((i · 𝐴) ∈ ℝ+ → 0 = (i · 𝐴))) |
17 | rpne0 12221 | . . . . . 6 ⊢ ((i · 𝐴) ∈ ℝ+ → (i · 𝐴) ≠ 0) | |
18 | 17 | necon2bi 2992 | . . . . 5 ⊢ ((i · 𝐴) = 0 → ¬ (i · 𝐴) ∈ ℝ+) |
19 | 18 | eqcoms 2781 | . . . 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 182 | . 2 ⊢ (𝐴 ∈ ℝ → ¬ (i · 𝐴) ∈ ℝ+) |
22 | df-nel 3069 | . 2 ⊢ ((i · 𝐴) ∉ ℝ+ ↔ ¬ (i · 𝐴) ∈ ℝ+) | |
23 | 21, 22 | sylibr 226 | 1 ⊢ (𝐴 ∈ ℝ → (i · 𝐴) ∉ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1508 ∈ wcel 2051 ∉ wnel 3068 ‘cfv 6186 (class class class)co 6975 ℂcc 10332 ℝcr 10333 0cc0 10334 ici 10336 + caddc 10337 · cmul 10339 ℝ+crp 12203 ℜcre 14316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-po 5323 df-so 5324 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-2 11502 df-rp 12204 df-cj 14318 df-re 14319 df-im 14320 |
This theorem is referenced by: sqr0lem 14460 resqreu 14472 resqrtcl 14473 |
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