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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 14457 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) | |
2 | cjth 14454 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ)) | |
3 | 2 | simpld 498 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) ∈ ℝ) |
4 | 3 | rehalfcld 11872 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) / 2) ∈ ℝ) |
5 | 1, 4 | eqeltrd 2890 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 ici 10528 + caddc 10529 · cmul 10531 − cmin 10859 / cdiv 11286 2c2 11680 ∗ccj 14447 ℜcre 14448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 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 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 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 11688 df-cj 14450 df-re 14451 |
This theorem is referenced by: imcl 14462 ref 14463 crre 14465 remim 14468 reim0b 14470 rereb 14471 mulre 14472 cjreb 14474 recj 14475 reneg 14476 readd 14477 resub 14478 remullem 14479 remul2 14481 rediv 14482 imcj 14483 imneg 14484 imadd 14485 immul2 14488 cjadd 14492 ipcnval 14494 cjmulval 14496 cjmulge0 14497 cjneg 14498 imval2 14502 cnrecnv 14516 sqeqd 14517 recli 14518 recld 14545 cnpart 14591 absrele 14660 releabs 14673 efeul 15507 absef 15542 absefib 15543 efieq1re 15544 cnsubrg 20151 mbfconst 24237 itgconst 24422 tanregt0 25131 argregt0 25201 tanarg 25210 logf1o2 25241 abscxp 25283 isosctrlem1 25404 asinsin 25478 acoscos 25479 atancj 25496 atantan 25509 cxploglim2 25564 zetacvg 25600 cncph 28602 ccfldextdgrr 31145 sqrtcvallem2 40337 sqrtcvallem3 40338 sqrtcvallem4 40339 sqrtcvallem5 40340 sqrtcval 40341 |
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