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Mirrors > Home > ILE Home > Th. List > recl | 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 | โข (๐ด โ โ โ (โโ๐ด) โ โ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reval 10825 | . 2 โข (๐ด โ โ โ (โโ๐ด) = ((๐ด + (โโ๐ด)) / 2)) | |
2 | cjth 10822 | . . . 4 โข (๐ด โ โ โ ((๐ด + (โโ๐ด)) โ โ โง (i ยท (๐ด โ (โโ๐ด))) โ โ)) | |
3 | 2 | simpld 112 | . . 3 โข (๐ด โ โ โ (๐ด + (โโ๐ด)) โ โ) |
4 | 3 | rehalfcld 9136 | . 2 โข (๐ด โ โ โ ((๐ด + (โโ๐ด)) / 2) โ โ) |
5 | 1, 4 | eqeltrd 2252 | 1 โข (๐ด โ โ โ (โโ๐ด) โ โ) |
Colors of variables: wff set class |
Syntax hints: โ wi 4 โ wcel 2146 โcfv 5208 (class class class)co 5865 โcc 7784 โcr 7785 ici 7788 + caddc 7789 ยท cmul 7791 โ cmin 8102 / cdiv 8601 2c2 8941 โccj 10815 โcre 10816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-po 4290 df-iso 4291 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8602 df-2 8949 df-cj 10818 df-re 10819 |
This theorem is referenced by: imcl 10830 ref 10831 crre 10833 remim 10836 reim0b 10838 rereb 10839 mulreap 10840 cjreb 10842 recj 10843 reneg 10844 readd 10845 resub 10846 remullem 10847 remul2 10849 redivap 10850 imcj 10851 imneg 10852 imadd 10853 immul2 10856 cjadd 10860 ipcnval 10862 cjmulval 10864 cjmulge0 10865 cjneg 10866 imval2 10870 cnrecnv 10886 recli 10887 recld 10914 cnreim 10954 abs00ap 11038 absrele 11059 releabs 11072 efeul 11709 absef 11744 absefib 11745 efieq1re 11746 abscxp 13828 |
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