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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 15153 | . 2 ⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) | |
2 | replim 15151 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | |
3 | 2 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) |
4 | oveq2 7438 | . . . . . . . 8 ⊢ ((ℑ‘𝐴) = 0 → (i · (ℑ‘𝐴)) = (i · 0)) | |
5 | it0e0 12485 | . . . . . . . 8 ⊢ (i · 0) = 0 | |
6 | 4, 5 | eqtrdi 2790 | . . . . . . 7 ⊢ ((ℑ‘𝐴) = 0 → (i · (ℑ‘𝐴)) = 0) |
7 | 6 | oveq2d 7446 | . . . . . 6 ⊢ ((ℑ‘𝐴) = 0 → ((ℜ‘𝐴) + (i · (ℑ‘𝐴))) = ((ℜ‘𝐴) + 0)) |
8 | recl 15145 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
9 | 8 | recnd 11286 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
10 | 9 | addridd 11458 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) + 0) = (ℜ‘𝐴)) |
11 | 7, 10 | sylan9eqr 2796 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → ((ℜ‘𝐴) + (i · (ℑ‘𝐴))) = (ℜ‘𝐴)) |
12 | 3, 11 | eqtrd 2774 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → 𝐴 = (ℜ‘𝐴)) |
13 | 8 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → (ℜ‘𝐴) ∈ ℝ) |
14 | 12, 13 | eqeltrd 2838 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → 𝐴 ∈ ℝ) |
15 | 14 | ex 412 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℑ‘𝐴) = 0 → 𝐴 ∈ ℝ)) |
16 | 1, 15 | impbid2 226 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 ici 11154 + caddc 11155 · cmul 11157 ℜcre 15132 ℑcim 15133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-2 12326 df-cj 15134 df-re 15135 df-im 15136 |
This theorem is referenced by: cjreb 15158 reim0bi 15207 reim0bd 15235 cnpart 15275 rlimrecl 15612 absefib 16230 efieq1re 16231 cnsubrg 21462 recld2 24849 aaliou2b 26397 logcj 26662 argimgt0 26668 logcnlem2 26699 logcnlem3 26700 logf1o2 26706 constrrtll 33736 sqrtcvallem1 43620 dstregt0 45231 absimnre 45426 readdcnnred 47252 resubcnnred 47253 cndivrenred 47255 |
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