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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 9172 | . . . 4 | |
2 | mulcom 7861 | . . . 4 | |
3 | 1, 2 | sylan2 284 | . . 3 |
4 | 3 | fveq2d 5472 | . 2 |
5 | oveq2 5832 | . . . . . 6 | |
6 | 5 | fveq2d 5472 | . . . . 5 |
7 | oveq2 5832 | . . . . 5 | |
8 | 6, 7 | eqeq12d 2172 | . . . 4 |
9 | oveq2 5832 | . . . . . 6 | |
10 | 9 | fveq2d 5472 | . . . . 5 |
11 | oveq2 5832 | . . . . 5 | |
12 | 10, 11 | eqeq12d 2172 | . . . 4 |
13 | oveq2 5832 | . . . . . 6 | |
14 | 13 | fveq2d 5472 | . . . . 5 |
15 | oveq2 5832 | . . . . 5 | |
16 | 14, 15 | eqeq12d 2172 | . . . 4 |
17 | oveq2 5832 | . . . . . 6 | |
18 | 17 | fveq2d 5472 | . . . . 5 |
19 | oveq2 5832 | . . . . 5 | |
20 | 18, 19 | eqeq12d 2172 | . . . 4 |
21 | oveq2 5832 | . . . . . 6 | |
22 | 21 | fveq2d 5472 | . . . . 5 |
23 | oveq2 5832 | . . . . 5 | |
24 | 22, 23 | eqeq12d 2172 | . . . 4 |
25 | ef0 11569 | . . . . 5 | |
26 | mul01 8264 | . . . . . 6 | |
27 | 26 | fveq2d 5472 | . . . . 5 |
28 | efcl 11561 | . . . . . 6 | |
29 | 28 | exp0d 10545 | . . . . 5 |
30 | 25, 27, 29 | 3eqtr4a 2216 | . . . 4 |
31 | oveq1 5831 | . . . . . . 7 | |
32 | 31 | adantl 275 | . . . . . 6 |
33 | nn0cn 9100 | . . . . . . . . . 10 | |
34 | ax-1cn 7825 | . . . . . . . . . . . 12 | |
35 | adddi 7864 | . . . . . . . . . . . 12 | |
36 | 34, 35 | mp3an3 1308 | . . . . . . . . . . 11 |
37 | mulid1 7875 | . . . . . . . . . . . . 13 | |
38 | 37 | adantr 274 | . . . . . . . . . . . 12 |
39 | 38 | oveq2d 5840 | . . . . . . . . . . 11 |
40 | 36, 39 | eqtrd 2190 | . . . . . . . . . 10 |
41 | 33, 40 | sylan2 284 | . . . . . . . . 9 |
42 | 41 | fveq2d 5472 | . . . . . . . 8 |
43 | mulcl 7859 | . . . . . . . . . 10 | |
44 | 33, 43 | sylan2 284 | . . . . . . . . 9 |
45 | simpl 108 | . . . . . . . . 9 | |
46 | efadd 11572 | . . . . . . . . 9 | |
47 | 44, 45, 46 | syl2anc 409 | . . . . . . . 8 |
48 | 42, 47 | eqtrd 2190 | . . . . . . 7 |
49 | 48 | adantr 274 | . . . . . 6 |
50 | expp1 10426 | . . . . . . . 8 | |
51 | 28, 50 | sylan 281 | . . . . . . 7 |
52 | 51 | adantr 274 | . . . . . 6 |
53 | 32, 49, 52 | 3eqtr4d 2200 | . . . . 5 |
54 | 53 | exp31 362 | . . . 4 |
55 | oveq2 5832 | . . . . . 6 | |
56 | nncn 8841 | . . . . . . . . . 10 | |
57 | mulneg2 8271 | . . . . . . . . . 10 | |
58 | 56, 57 | sylan2 284 | . . . . . . . . 9 |
59 | 58 | fveq2d 5472 | . . . . . . . 8 |
60 | 56, 43 | sylan2 284 | . . . . . . . . 9 |
61 | efneg 11576 | . . . . . . . . 9 | |
62 | 60, 61 | syl 14 | . . . . . . . 8 |
63 | 59, 62 | eqtrd 2190 | . . . . . . 7 |
64 | efap0 11574 | . . . . . . . 8 # | |
65 | nnnn0 9097 | . . . . . . . 8 | |
66 | expnegap0 10427 | . . . . . . . 8 # | |
67 | 28, 64, 65, 66 | syl2an3an 1280 | . . . . . . 7 |
68 | 63, 67 | eqeq12d 2172 | . . . . . 6 |
69 | 55, 68 | syl5ibr 155 | . . . . 5 |
70 | 69 | ex 114 | . . . 4 |
71 | 8, 12, 16, 20, 24, 30, 54, 70 | zindd 9282 | . . 3 |
72 | 71 | imp 123 | . 2 |
73 | 4, 72 | eqtr3d 2192 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 class class class wbr 3965 cfv 5170 (class class class)co 5824 cc 7730 cc0 7732 c1 7733 caddc 7735 cmul 7737 cneg 8047 # cap 8456 cdiv 8545 cn 8833 cn0 9090 cz 9167 cexp 10418 ce 11539 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-disj 3943 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-isom 5179 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-frec 6338 df-1o 6363 df-oadd 6367 df-er 6480 df-en 6686 df-dom 6687 df-fin 6688 df-sup 6928 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-n0 9091 df-z 9168 df-uz 9440 df-q 9529 df-rp 9561 df-ico 9798 df-fz 9913 df-fzo 10042 df-seqfrec 10345 df-exp 10419 df-fac 10600 df-bc 10622 df-ihash 10650 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899 df-clim 11176 df-sumdc 11251 df-ef 11545 |
This theorem is referenced by: efzval 11580 efgt0 11581 tanval3ap 11611 demoivre 11669 ef2kpi 13138 reexplog 13203 relogexp 13204 |
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