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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 10616 | . 2 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(-i · 𝐴))) | |
2 | negicn 7956 | . . . 4 ⊢ -i ∈ ℂ | |
3 | mulcl 7740 | . . . 4 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) ∈ ℂ) | |
4 | 2, 3 | mpan 420 | . . 3 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
5 | recl 10618 | . . 3 ⊢ ((-i · 𝐴) ∈ ℂ → (ℜ‘(-i · 𝐴)) ∈ ℝ) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘(-i · 𝐴)) ∈ ℝ) |
7 | 1, 6 | eqeltrd 2214 | 1 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ‘cfv 5118 (class class class)co 5767 ℂcc 7611 ℝcr 7612 ici 7615 · cmul 7618 -cneg 7927 ℜcre 10605 ℑcim 10606 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-2 8772 df-cj 10607 df-re 10608 df-im 10609 |
This theorem is referenced by: imf 10621 remim 10625 mulreap 10629 cjreb 10631 recj 10632 reneg 10633 readd 10634 remullem 10636 remul2 10638 imcj 10640 imneg 10641 imadd 10642 imsub 10643 immul2 10645 imdivap 10646 cjcj 10648 cjadd 10649 ipcnval 10651 cjmulval 10653 cjmulge0 10654 cjneg 10655 imval2 10659 cnrecnv 10675 imcli 10677 imcld 10704 cnreim 10743 abs00ap 10827 absrele 10848 efeul 11430 absef 11465 absefib 11466 efieq1re 11467 |
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