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Mirrors > Home > MPE Home > Th. List > recl | Structured version Visualization version GIF version |
Description: The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
recl | ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reval 14455 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) | |
2 | cjth 14452 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ)) | |
3 | 2 | simpld 495 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) ∈ ℝ) |
4 | 3 | rehalfcld 11873 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) / 2) ∈ ℝ) |
5 | 1, 4 | eqeltrd 2913 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 ℝcr 10525 ici 10528 + caddc 10529 · cmul 10531 − cmin 10859 / cdiv 11286 2c2 11681 ∗ccj 14445 ℜcre 14446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-2 11689 df-cj 14448 df-re 14449 |
This theorem is referenced by: imcl 14460 ref 14461 crre 14463 remim 14466 reim0b 14468 rereb 14469 mulre 14470 cjreb 14472 recj 14473 reneg 14474 readd 14475 resub 14476 remullem 14477 remul2 14479 rediv 14480 imcj 14481 imneg 14482 imadd 14483 immul2 14486 cjadd 14490 ipcnval 14492 cjmulval 14494 cjmulge0 14495 cjneg 14496 imval2 14500 cnrecnv 14514 sqeqd 14515 recli 14516 recld 14543 cnpart 14589 absrele 14658 releabs 14671 efeul 15505 absef 15540 absefib 15541 efieq1re 15542 cnsubrg 20535 mbfconst 24163 itgconst 24348 tanregt0 25050 argregt0 25120 tanarg 25129 logf1o2 25160 abscxp 25202 isosctrlem1 25323 asinsin 25397 acoscos 25398 atancj 25415 atantan 25428 cxploglim2 25484 zetacvg 25520 cncph 28524 ccfldextdgrr 30957 |
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