![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > imcl | Structured version Visualization version 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 14289 | . 2 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(-i · 𝐴))) | |
2 | negicn 10723 | . . . 4 ⊢ -i ∈ ℂ | |
3 | mulcl 10456 | . . . 4 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) ∈ ℂ) | |
4 | 2, 3 | mpan 686 | . . 3 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
5 | recl 14291 | . . 3 ⊢ ((-i · 𝐴) ∈ ℂ → (ℜ‘(-i · 𝐴)) ∈ ℝ) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘(-i · 𝐴)) ∈ ℝ) |
7 | 1, 6 | eqeltrd 2881 | 1 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2079 ‘cfv 6217 (class class class)co 7007 ℂcc 10370 ℝcr 10371 ici 10374 · cmul 10377 -cneg 10707 ℜcre 14278 ℑcim 14279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-po 5354 df-so 5355 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-2 11537 df-cj 14280 df-re 14281 df-im 14282 |
This theorem is referenced by: imf 14294 remim 14298 mulre 14302 cjreb 14304 recj 14305 reneg 14306 readd 14307 remullem 14309 remul2 14311 imcj 14313 imneg 14314 imadd 14315 imsub 14316 immul2 14318 imdiv 14319 cjcj 14321 cjadd 14322 ipcnval 14324 cjmulval 14326 cjmulge0 14327 cjneg 14328 imval2 14332 cnrecnv 14346 imcli 14349 imcld 14376 absrele 14490 efeul 15336 absef 15371 absefib 15372 efieq1re 15373 cnsubrg 20275 mbfconst 23905 itgconst 24090 tanregt0 24792 ellogrn 24812 argimgt0 24864 argimlt0 24865 logneg2 24867 tanarg 24871 logf1o2 24902 logreclem 25009 asinlem3a 25117 asinlem3 25118 zetacvg 25262 ccfldextdgrr 30616 sigarls 42610 |
Copyright terms: Public domain | W3C validator |