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Mirrors > Home > ILE Home > Th. List > efexp | Unicode version |
Description: The exponential of an integer power. Corollary 15-4.4 of [Gleason] p. 309, restricted to integers. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 5-Jun-2014.) |
Ref | Expression |
---|---|
efexp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9059 | . . . 4 | |
2 | mulcom 7749 | . . . 4 | |
3 | 1, 2 | sylan2 284 | . . 3 |
4 | 3 | fveq2d 5425 | . 2 |
5 | oveq2 5782 | . . . . . 6 | |
6 | 5 | fveq2d 5425 | . . . . 5 |
7 | oveq2 5782 | . . . . 5 | |
8 | 6, 7 | eqeq12d 2154 | . . . 4 |
9 | oveq2 5782 | . . . . . 6 | |
10 | 9 | fveq2d 5425 | . . . . 5 |
11 | oveq2 5782 | . . . . 5 | |
12 | 10, 11 | eqeq12d 2154 | . . . 4 |
13 | oveq2 5782 | . . . . . 6 | |
14 | 13 | fveq2d 5425 | . . . . 5 |
15 | oveq2 5782 | . . . . 5 | |
16 | 14, 15 | eqeq12d 2154 | . . . 4 |
17 | oveq2 5782 | . . . . . 6 | |
18 | 17 | fveq2d 5425 | . . . . 5 |
19 | oveq2 5782 | . . . . 5 | |
20 | 18, 19 | eqeq12d 2154 | . . . 4 |
21 | oveq2 5782 | . . . . . 6 | |
22 | 21 | fveq2d 5425 | . . . . 5 |
23 | oveq2 5782 | . . . . 5 | |
24 | 22, 23 | eqeq12d 2154 | . . . 4 |
25 | ef0 11378 | . . . . 5 | |
26 | mul01 8151 | . . . . . 6 | |
27 | 26 | fveq2d 5425 | . . . . 5 |
28 | efcl 11370 | . . . . . 6 | |
29 | 28 | exp0d 10418 | . . . . 5 |
30 | 25, 27, 29 | 3eqtr4a 2198 | . . . 4 |
31 | oveq1 5781 | . . . . . . 7 | |
32 | 31 | adantl 275 | . . . . . 6 |
33 | nn0cn 8987 | . . . . . . . . . 10 | |
34 | ax-1cn 7713 | . . . . . . . . . . . 12 | |
35 | adddi 7752 | . . . . . . . . . . . 12 | |
36 | 34, 35 | mp3an3 1304 | . . . . . . . . . . 11 |
37 | mulid1 7763 | . . . . . . . . . . . . 13 | |
38 | 37 | adantr 274 | . . . . . . . . . . . 12 |
39 | 38 | oveq2d 5790 | . . . . . . . . . . 11 |
40 | 36, 39 | eqtrd 2172 | . . . . . . . . . 10 |
41 | 33, 40 | sylan2 284 | . . . . . . . . 9 |
42 | 41 | fveq2d 5425 | . . . . . . . 8 |
43 | mulcl 7747 | . . . . . . . . . 10 | |
44 | 33, 43 | sylan2 284 | . . . . . . . . 9 |
45 | simpl 108 | . . . . . . . . 9 | |
46 | efadd 11381 | . . . . . . . . 9 | |
47 | 44, 45, 46 | syl2anc 408 | . . . . . . . 8 |
48 | 42, 47 | eqtrd 2172 | . . . . . . 7 |
49 | 48 | adantr 274 | . . . . . 6 |
50 | expp1 10300 | . . . . . . . 8 | |
51 | 28, 50 | sylan 281 | . . . . . . 7 |
52 | 51 | adantr 274 | . . . . . 6 |
53 | 32, 49, 52 | 3eqtr4d 2182 | . . . . 5 |
54 | 53 | exp31 361 | . . . 4 |
55 | oveq2 5782 | . . . . . 6 | |
56 | nncn 8728 | . . . . . . . . . 10 | |
57 | mulneg2 8158 | . . . . . . . . . 10 | |
58 | 56, 57 | sylan2 284 | . . . . . . . . 9 |
59 | 58 | fveq2d 5425 | . . . . . . . 8 |
60 | 56, 43 | sylan2 284 | . . . . . . . . 9 |
61 | efneg 11385 | . . . . . . . . 9 | |
62 | 60, 61 | syl 14 | . . . . . . . 8 |
63 | 59, 62 | eqtrd 2172 | . . . . . . 7 |
64 | efap0 11383 | . . . . . . . 8 # | |
65 | nnnn0 8984 | . . . . . . . 8 | |
66 | expnegap0 10301 | . . . . . . . 8 # | |
67 | 28, 64, 65, 66 | syl2an3an 1276 | . . . . . . 7 |
68 | 63, 67 | eqeq12d 2154 | . . . . . 6 |
69 | 55, 68 | syl5ibr 155 | . . . . 5 |
70 | 69 | ex 114 | . . . 4 |
71 | 8, 12, 16, 20, 24, 30, 54, 70 | zindd 9169 | . . 3 |
72 | 71 | imp 123 | . 2 |
73 | 4, 72 | eqtr3d 2174 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc 7618 cc0 7620 c1 7621 caddc 7623 cmul 7625 cneg 7934 # cap 8343 cdiv 8432 cn 8720 cn0 8977 cz 9054 cexp 10292 ce 11348 |
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 ax-arch 7739 ax-caucvg 7740 |
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-disj 3907 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-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-sup 6871 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-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-ico 9677 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-fac 10472 df-bc 10494 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 df-ef 11354 |
This theorem is referenced by: efzval 11389 efgt0 11390 tanval3ap 11421 demoivre 11479 ef2kpi 12887 |
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