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Mirrors > Home > ILE Home > Th. List > cjexp | Unicode version |
Description: Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.) |
Ref | Expression |
---|---|
cjexp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5782 | . . . . . 6 | |
2 | 1 | fveq2d 5425 | . . . . 5 |
3 | oveq2 5782 | . . . . 5 | |
4 | 2, 3 | eqeq12d 2154 | . . . 4 |
5 | 4 | imbi2d 229 | . . 3 |
6 | oveq2 5782 | . . . . . 6 | |
7 | 6 | fveq2d 5425 | . . . . 5 |
8 | oveq2 5782 | . . . . 5 | |
9 | 7, 8 | eqeq12d 2154 | . . . 4 |
10 | 9 | imbi2d 229 | . . 3 |
11 | oveq2 5782 | . . . . . 6 | |
12 | 11 | fveq2d 5425 | . . . . 5 |
13 | oveq2 5782 | . . . . 5 | |
14 | 12, 13 | eqeq12d 2154 | . . . 4 |
15 | 14 | imbi2d 229 | . . 3 |
16 | oveq2 5782 | . . . . . 6 | |
17 | 16 | fveq2d 5425 | . . . . 5 |
18 | oveq2 5782 | . . . . 5 | |
19 | 17, 18 | eqeq12d 2154 | . . . 4 |
20 | 19 | imbi2d 229 | . . 3 |
21 | exp0 10297 | . . . . 5 | |
22 | 21 | fveq2d 5425 | . . . 4 |
23 | cjcl 10620 | . . . . 5 | |
24 | exp0 10297 | . . . . . 6 | |
25 | 1re 7765 | . . . . . . 7 | |
26 | cjre 10654 | . . . . . . 7 | |
27 | 25, 26 | ax-mp 5 | . . . . . 6 |
28 | 24, 27 | syl6eqr 2190 | . . . . 5 |
29 | 23, 28 | syl 14 | . . . 4 |
30 | 22, 29 | eqtr4d 2175 | . . 3 |
31 | expp1 10300 | . . . . . . . . . 10 | |
32 | 31 | fveq2d 5425 | . . . . . . . . 9 |
33 | expcl 10311 | . . . . . . . . . 10 | |
34 | simpl 108 | . . . . . . . . . 10 | |
35 | cjmul 10657 | . . . . . . . . . 10 | |
36 | 33, 34, 35 | syl2anc 408 | . . . . . . . . 9 |
37 | 32, 36 | eqtrd 2172 | . . . . . . . 8 |
38 | 37 | adantr 274 | . . . . . . 7 |
39 | oveq1 5781 | . . . . . . . 8 | |
40 | expp1 10300 | . . . . . . . . . 10 | |
41 | 23, 40 | sylan 281 | . . . . . . . . 9 |
42 | 41 | eqcomd 2145 | . . . . . . . 8 |
43 | 39, 42 | sylan9eqr 2194 | . . . . . . 7 |
44 | 38, 43 | eqtrd 2172 | . . . . . 6 |
45 | 44 | exp31 361 | . . . . 5 |
46 | 45 | com12 30 | . . . 4 |
47 | 46 | a2d 26 | . . 3 |
48 | 5, 10, 15, 20, 30, 47 | nn0ind 9165 | . 2 |
49 | 48 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cfv 5123 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 caddc 7623 cmul 7625 cn0 8977 cexp 10292 ccj 10611 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 |
This theorem is referenced by: cjexpd 10730 efcj 11379 |
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