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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 14555 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = ((𝐴 + (∗‘𝐴)) / 2)) | |
2 | cjth 14552 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) ∈ ℝ ∧ (i · (𝐴 − (∗‘𝐴))) ∈ ℝ)) | |
3 | 2 | simpld 498 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 + (∗‘𝐴)) ∈ ℝ) |
4 | 3 | rehalfcld 11963 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + (∗‘𝐴)) / 2) ∈ ℝ) |
5 | 1, 4 | eqeltrd 2833 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ‘cfv 6339 (class class class)co 7170 ℂcc 10613 ℝcr 10614 ici 10617 + caddc 10618 · cmul 10620 − cmin 10948 / cdiv 11375 2c2 11771 ∗ccj 14545 ℜcre 14546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-2 11779 df-cj 14548 df-re 14549 |
This theorem is referenced by: imcl 14560 ref 14561 crre 14563 remim 14566 reim0b 14568 rereb 14569 mulre 14570 cjreb 14572 recj 14573 reneg 14574 readd 14575 resub 14576 remullem 14577 remul2 14579 rediv 14580 imcj 14581 imneg 14582 imadd 14583 immul2 14586 cjadd 14590 ipcnval 14592 cjmulval 14594 cjmulge0 14595 cjneg 14596 imval2 14600 cnrecnv 14614 sqeqd 14615 recli 14616 recld 14643 cnpart 14689 absrele 14758 releabs 14771 efeul 15607 absef 15642 absefib 15643 efieq1re 15644 cnsubrg 20277 mbfconst 24385 itgconst 24571 tanregt0 25283 argregt0 25353 tanarg 25362 logf1o2 25393 abscxp 25435 isosctrlem1 25556 asinsin 25630 acoscos 25631 atancj 25648 atantan 25661 cxploglim2 25716 zetacvg 25752 cncph 28754 ccfldextdgrr 31314 sqrtcvallem2 40790 sqrtcvallem3 40791 sqrtcvallem4 40792 sqrtcvallem5 40793 sqrtcval 40794 |
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