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Mirrors > Home > ILE Home > Th. List > iexpcyc | Unicode version |
Description: Taking to the -th power is the same as using the -th power instead, by i4 10530. (Contributed by Mario Carneiro, 7-Jul-2014.) |
Ref | Expression |
---|---|
iexpcyc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zq 9541 | . . . 4 | |
2 | 4z 9202 | . . . . . 6 | |
3 | zq 9541 | . . . . . 6 | |
4 | 2, 3 | ax-mp 5 | . . . . 5 |
5 | 4pos 8935 | . . . . 5 | |
6 | modqval 10232 | . . . . 5 | |
7 | 4, 5, 6 | mp3an23 1311 | . . . 4 |
8 | 1, 7 | syl 14 | . . 3 |
9 | 8 | oveq2d 5842 | . 2 |
10 | 4nn 9001 | . . . . . . 7 | |
11 | znq 9539 | . . . . . . 7 | |
12 | 10, 11 | mpan2 422 | . . . . . 6 |
13 | 12 | flqcld 10185 | . . . . 5 |
14 | zmulcl 9225 | . . . . 5 | |
15 | 2, 13, 14 | sylancr 411 | . . . 4 |
16 | ax-icn 7829 | . . . . 5 | |
17 | iap0 9061 | . . . . 5 # | |
18 | expsubap 10476 | . . . . 5 # | |
19 | 16, 17, 18 | mpanl12 433 | . . . 4 |
20 | 15, 19 | mpdan 418 | . . 3 |
21 | expmulzap 10474 | . . . . . . . 8 # | |
22 | 16, 17, 21 | mpanl12 433 | . . . . . . 7 |
23 | 2, 13, 22 | sylancr 411 | . . . . . 6 |
24 | i4 10530 | . . . . . . . 8 | |
25 | 24 | oveq1i 5836 | . . . . . . 7 |
26 | 1exp 10457 | . . . . . . . 8 | |
27 | 13, 26 | syl 14 | . . . . . . 7 |
28 | 25, 27 | syl5eq 2202 | . . . . . 6 |
29 | 23, 28 | eqtrd 2190 | . . . . 5 |
30 | 29 | oveq2d 5842 | . . . 4 |
31 | expclzap 10453 | . . . . . 6 # | |
32 | 16, 17, 31 | mp3an12 1309 | . . . . 5 |
33 | 32 | div1d 8657 | . . . 4 |
34 | 30, 33 | eqtrd 2190 | . . 3 |
35 | 20, 34 | eqtrd 2190 | . 2 |
36 | 9, 35 | eqtrd 2190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 class class class wbr 3967 cfv 5172 (class class class)co 5826 cc 7732 cc0 7734 c1 7735 ci 7736 cmul 7739 clt 7914 cmin 8050 # cap 8460 cdiv 8549 cn 8838 c4 8891 cz 9172 cq 9534 cfl 10176 cmo 10230 cexp 10427 |
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 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 ax-pre-mulext 7852 ax-arch 7853 |
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 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-id 4255 df-po 4258 df-iso 4259 df-iord 4328 df-on 4330 df-ilim 4331 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-frec 6340 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-div 8550 df-inn 8839 df-2 8897 df-3 8898 df-4 8899 df-n0 9096 df-z 9173 df-uz 9445 df-q 9535 df-rp 9567 df-fl 10178 df-mod 10231 df-seqfrec 10354 df-exp 10428 |
This theorem is referenced by: (None) |
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