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Mirrors > Home > ILE Home > Th. List > imcl | GIF version |
Description: The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
imcl | ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imre 10755 | . 2 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(-i · 𝐴))) | |
2 | negicn 8077 | . . . 4 ⊢ -i ∈ ℂ | |
3 | mulcl 7860 | . . . 4 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) ∈ ℂ) | |
4 | 2, 3 | mpan 421 | . . 3 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
5 | recl 10757 | . . 3 ⊢ ((-i · 𝐴) ∈ ℂ → (ℜ‘(-i · 𝐴)) ∈ ℝ) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘(-i · 𝐴)) ∈ ℝ) |
7 | 1, 6 | eqeltrd 2234 | 1 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2128 ‘cfv 5171 (class class class)co 5825 ℂcc 7731 ℝcr 7732 ici 7735 · cmul 7738 -cneg 8048 ℜcre 10744 ℑcim 10745 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-po 4257 df-iso 4258 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-2 8893 df-cj 10746 df-re 10747 df-im 10748 |
This theorem is referenced by: imf 10760 remim 10764 mulreap 10768 cjreb 10770 recj 10771 reneg 10772 readd 10773 remullem 10775 remul2 10777 imcj 10779 imneg 10780 imadd 10781 imsub 10782 immul2 10784 imdivap 10785 cjcj 10787 cjadd 10788 ipcnval 10790 cjmulval 10792 cjmulge0 10793 cjneg 10794 imval2 10798 cnrecnv 10814 imcli 10816 imcld 10843 cnreim 10882 abs00ap 10966 absrele 10987 efeul 11635 absef 11670 absefib 11671 efieq1re 11672 |
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