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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 14480 | . 2 ⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) | |
2 | replim 14478 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | |
3 | 2 | adantr 483 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) |
4 | oveq2 7167 | . . . . . . . 8 ⊢ ((ℑ‘𝐴) = 0 → (i · (ℑ‘𝐴)) = (i · 0)) | |
5 | it0e0 11862 | . . . . . . . 8 ⊢ (i · 0) = 0 | |
6 | 4, 5 | syl6eq 2875 | . . . . . . 7 ⊢ ((ℑ‘𝐴) = 0 → (i · (ℑ‘𝐴)) = 0) |
7 | 6 | oveq2d 7175 | . . . . . 6 ⊢ ((ℑ‘𝐴) = 0 → ((ℜ‘𝐴) + (i · (ℑ‘𝐴))) = ((ℜ‘𝐴) + 0)) |
8 | recl 14472 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
9 | 8 | recnd 10672 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
10 | 9 | addid1d 10843 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) + 0) = (ℜ‘𝐴)) |
11 | 7, 10 | sylan9eqr 2881 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → ((ℜ‘𝐴) + (i · (ℑ‘𝐴))) = (ℜ‘𝐴)) |
12 | 3, 11 | eqtrd 2859 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → 𝐴 = (ℜ‘𝐴)) |
13 | 8 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → (ℜ‘𝐴) ∈ ℝ) |
14 | 12, 13 | eqeltrd 2916 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) = 0) → 𝐴 ∈ ℝ) |
15 | 14 | ex 415 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℑ‘𝐴) = 0 → 𝐴 ∈ ℝ)) |
16 | 1, 15 | impbid2 228 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 ici 10542 + caddc 10543 · cmul 10545 ℜcre 14459 ℑcim 14460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-2 11703 df-cj 14461 df-re 14462 df-im 14463 |
This theorem is referenced by: cjreb 14485 reim0bi 14534 reim0bd 14562 cnpart 14602 rlimrecl 14940 absefib 15554 efieq1re 15555 cnsubrg 20608 recld2 23425 aaliou2b 24933 logcj 25192 argimgt0 25198 logcnlem2 25229 logcnlem3 25230 logf1o2 25236 dstregt0 41553 absimnre 41759 readdcnnred 43510 resubcnnred 43511 cndivrenred 43513 |
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