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Mirrors > Home > ILE Home > Th. List > absefib | Unicode version |
Description: A complex number is real iff the exponential of its product with has absolute value one. (Contributed by NM, 21-Aug-2008.) |
Ref | Expression |
---|---|
absefib |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ef0 11622 | . . . . 5 | |
2 | 1 | eqeq2i 2181 | . . . 4 |
3 | imcl 10805 | . . . . . 6 | |
4 | 3 | renegcld 8286 | . . . . 5 |
5 | 0re 7907 | . . . . 5 | |
6 | reef11 11649 | . . . . 5 | |
7 | 4, 5, 6 | sylancl 411 | . . . 4 |
8 | 2, 7 | bitr3id 193 | . . 3 |
9 | 3 | recnd 7935 | . . . 4 |
10 | 9 | negeq0d 8209 | . . 3 |
11 | 8, 10 | bitr4d 190 | . 2 |
12 | ax-icn 7856 | . . . . . 6 | |
13 | mulcl 7888 | . . . . . 6 | |
14 | 12, 13 | mpan 422 | . . . . 5 |
15 | absef 11719 | . . . . 5 | |
16 | 14, 15 | syl 14 | . . . 4 |
17 | replim 10810 | . . . . . . . . . 10 | |
18 | recl 10804 | . . . . . . . . . . . 12 | |
19 | 18 | recnd 7935 | . . . . . . . . . . 11 |
20 | mulcl 7888 | . . . . . . . . . . . 12 | |
21 | 12, 9, 20 | sylancr 412 | . . . . . . . . . . 11 |
22 | 19, 21 | addcomd 8057 | . . . . . . . . . 10 |
23 | 17, 22 | eqtrd 2203 | . . . . . . . . 9 |
24 | 23 | oveq2d 5866 | . . . . . . . 8 |
25 | adddi 7893 | . . . . . . . . . . 11 | |
26 | 12, 25 | mp3an1 1319 | . . . . . . . . . 10 |
27 | 21, 19, 26 | syl2anc 409 | . . . . . . . . 9 |
28 | ixi 8489 | . . . . . . . . . . . 12 | |
29 | 28 | oveq1i 5860 | . . . . . . . . . . 11 |
30 | mulass 7892 | . . . . . . . . . . . . 13 | |
31 | 12, 12, 30 | mp3an12 1322 | . . . . . . . . . . . 12 |
32 | 9, 31 | syl 14 | . . . . . . . . . . 11 |
33 | 9 | mulm1d 8316 | . . . . . . . . . . 11 |
34 | 29, 32, 33 | 3eqtr3a 2227 | . . . . . . . . . 10 |
35 | 34 | oveq1d 5865 | . . . . . . . . 9 |
36 | 27, 35 | eqtrd 2203 | . . . . . . . 8 |
37 | 24, 36 | eqtrd 2203 | . . . . . . 7 |
38 | 37 | fveq2d 5498 | . . . . . 6 |
39 | 4, 18 | crred 10927 | . . . . . 6 |
40 | 38, 39 | eqtrd 2203 | . . . . 5 |
41 | 40 | fveq2d 5498 | . . . 4 |
42 | 16, 41 | eqtrd 2203 | . . 3 |
43 | 42 | eqeq1d 2179 | . 2 |
44 | reim0b 10813 | . 2 | |
45 | 11, 43, 44 | 3bitr4rd 220 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1348 wcel 2141 cfv 5196 (class class class)co 5850 cc 7759 cr 7760 cc0 7761 c1 7762 ci 7763 caddc 7764 cmul 7766 cneg 8078 cre 10791 cim 10792 cabs 10948 ce 11592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 ax-arch 7880 ax-caucvg 7881 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-disj 3965 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-frec 6367 df-1o 6392 df-oadd 6396 df-er 6509 df-en 6715 df-dom 6716 df-fin 6717 df-sup 6957 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-2 8924 df-3 8925 df-4 8926 df-n0 9123 df-z 9200 df-uz 9475 df-q 9566 df-rp 9598 df-ico 9838 df-fz 9953 df-fzo 10086 df-seqfrec 10389 df-exp 10463 df-fac 10647 df-bc 10669 df-ihash 10697 df-cj 10793 df-re 10794 df-im 10795 df-rsqrt 10949 df-abs 10950 df-clim 11229 df-sumdc 11304 df-ef 11598 df-sin 11600 df-cos 11601 |
This theorem is referenced by: (None) |
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