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Mirrors > Home > ILE Home > Th. List > efcj | Unicode version |
Description: The exponential of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.) |
Ref | Expression |
---|---|
efcj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjcl 10613 | . . 3 | |
2 | eqid 2137 | . . . 4 | |
3 | 2 | efcvg 11361 | . . 3 |
4 | 1, 3 | syl 14 | . 2 |
5 | nn0uz 9353 | . . 3 | |
6 | eqid 2137 | . . . 4 | |
7 | 6 | efcvg 11361 | . . 3 |
8 | seqex 10213 | . . . 4 | |
9 | 8 | a1i 9 | . . 3 |
10 | 0zd 9059 | . . 3 | |
11 | 6 | eftvalcn 11352 | . . . . . 6 |
12 | eftcl 11349 | . . . . . 6 | |
13 | 11, 12 | eqeltrd 2214 | . . . . 5 |
14 | 5, 10, 13 | serf 10240 | . . . 4 |
15 | 14 | ffvelrnda 5548 | . . 3 |
16 | addcl 7738 | . . . . . 6 | |
17 | 16 | adantl 275 | . . . . 5 |
18 | simpl 108 | . . . . . 6 | |
19 | elnn0uz 9356 | . . . . . . 7 | |
20 | 19 | biimpri 132 | . . . . . 6 |
21 | 18, 20, 13 | syl2an 287 | . . . . 5 |
22 | simpr 109 | . . . . . 6 | |
23 | 22, 5 | eleqtrdi 2230 | . . . . 5 |
24 | cjadd 10649 | . . . . . 6 | |
25 | 24 | adantl 275 | . . . . 5 |
26 | expcl 10304 | . . . . . . . . 9 | |
27 | faccl 10474 | . . . . . . . . . . 11 | |
28 | 27 | adantl 275 | . . . . . . . . . 10 |
29 | 28 | nncnd 8727 | . . . . . . . . 9 |
30 | 28 | nnap0d 8759 | . . . . . . . . 9 # |
31 | 26, 29, 30 | cjdivapd 10733 | . . . . . . . 8 |
32 | cjexp 10658 | . . . . . . . . 9 | |
33 | 28 | nnred 8726 | . . . . . . . . . 10 |
34 | 33 | cjred 10736 | . . . . . . . . 9 |
35 | 32, 34 | oveq12d 5785 | . . . . . . . 8 |
36 | 31, 35 | eqtrd 2170 | . . . . . . 7 |
37 | 11 | fveq2d 5418 | . . . . . . 7 |
38 | 2 | eftvalcn 11352 | . . . . . . . 8 |
39 | 1, 38 | sylan 281 | . . . . . . 7 |
40 | 36, 37, 39 | 3eqtr4d 2180 | . . . . . 6 |
41 | 18, 20, 40 | syl2an 287 | . . . . 5 |
42 | 20 | adantl 275 | . . . . . . 7 |
43 | 1 | ad2antrr 479 | . . . . . . . . 9 |
44 | 43, 42 | expcld 10417 | . . . . . . . 8 |
45 | 18, 20, 29 | syl2an 287 | . . . . . . . 8 |
46 | 18, 20, 30 | syl2an 287 | . . . . . . . 8 # |
47 | 44, 45, 46 | divclapd 8543 | . . . . . . 7 |
48 | oveq2 5775 | . . . . . . . . 9 | |
49 | fveq2 5414 | . . . . . . . . 9 | |
50 | 48, 49 | oveq12d 5785 | . . . . . . . 8 |
51 | 50, 2 | fvmptg 5490 | . . . . . . 7 |
52 | 42, 47, 51 | syl2anc 408 | . . . . . 6 |
53 | 52, 47 | eqeltrd 2214 | . . . . 5 |
54 | 17, 21, 23, 25, 41, 53, 17 | seq3homo 10276 | . . . 4 |
55 | 54 | eqcomd 2143 | . . 3 |
56 | 5, 7, 9, 10, 15, 55 | climcj 11083 | . 2 |
57 | climuni 11055 | . 2 | |
58 | 4, 56, 57 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2681 class class class wbr 3924 cmpt 3984 cfv 5118 (class class class)co 5767 cc 7611 cc0 7613 caddc 7616 # cap 8336 cdiv 8425 cn 8713 cn0 8970 cuz 9319 cseq 10211 cexp 10285 cfa 10464 ccj 10604 cli 11040 ce 11337 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-frec 6281 df-1o 6306 df-oadd 6310 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-ico 9670 df-fz 9784 df-fzo 9913 df-seqfrec 10212 df-exp 10286 df-fac 10465 df-ihash 10515 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-clim 11041 df-sumdc 11116 df-ef 11343 |
This theorem is referenced by: resinval 11411 recosval 11412 |
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