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Mirrors > Home > MPE Home > Th. List > reim0b | Structured version Visualization version GIF version |
Description: A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.) |
Ref | Expression |
---|---|
reim0b | ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reim0 14196 | . 2 ⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) | |
2 | replim 14194 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | |
3 | 2 | adantr 473 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) |
4 | oveq2 6884 | . . . . . . . 8 ⊢ ((ℑ‘𝐴) = 0 → (i · (ℑ‘𝐴)) = (i · 0)) | |
5 | it0e0 11538 | . . . . . . . 8 ⊢ (i · 0) = 0 | |
6 | 4, 5 | syl6eq 2847 | . . . . . . 7 ⊢ ((ℑ‘𝐴) = 0 → (i · (ℑ‘𝐴)) = 0) |
7 | 6 | oveq2d 6892 | . . . . . 6 ⊢ ((ℑ‘𝐴) = 0 → ((ℜ‘𝐴) + (i · (ℑ‘𝐴))) = ((ℜ‘𝐴) + 0)) |
8 | recl 14188 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
9 | 8 | recnd 10355 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
10 | 9 | addid1d 10524 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) + 0) = (ℜ‘𝐴)) |
11 | 7, 10 | sylan9eqr 2853 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → ((ℜ‘𝐴) + (i · (ℑ‘𝐴))) = (ℜ‘𝐴)) |
12 | 3, 11 | eqtrd 2831 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → 𝐴 = (ℜ‘𝐴)) |
13 | 8 | adantr 473 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → (ℜ‘𝐴) ∈ ℝ) |
14 | 12, 13 | eqeltrd 2876 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → 𝐴 ∈ ℝ) |
15 | 14 | ex 402 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℑ‘𝐴) = 0 → 𝐴 ∈ ℝ)) |
16 | 1, 15 | impbid2 218 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ‘cfv 6099 (class class class)co 6876 ℂcc 10220 ℝcr 10221 0cc0 10222 ici 10224 + caddc 10225 · cmul 10227 ℜcre 14175 ℑcim 14176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-po 5231 df-so 5232 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-2 11372 df-cj 14177 df-re 14178 df-im 14179 |
This theorem is referenced by: cjreb 14201 reim0bi 14250 reim0bd 14278 cnpart 14318 rlimrecl 14649 absefib 15261 efieq1re 15262 cnsubrg 20125 recld2 22942 aaliou2b 24434 logcj 24690 argimgt0 24696 logcnlem2 24727 logcnlem3 24728 logf1o2 24734 dstregt0 40227 absimnre 40438 |
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