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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 9577. (Contributed by Mario Carneiro, 7-Jul-2014.) |
Ref | Expression |
---|---|
iexpcyc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zq 8711 | . . . 4 | |
2 | 4z 8381 | . . . . . 6 | |
3 | zq 8711 | . . . . . 6 | |
4 | 2, 3 | ax-mp 7 | . . . . 5 |
5 | 4pos 8136 | . . . . 5 | |
6 | modqval 9326 | . . . . 5 | |
7 | 4, 5, 6 | mp3an23 1260 | . . . 4 |
8 | 1, 7 | syl 14 | . . 3 |
9 | 8 | oveq2d 5548 | . 2 |
10 | 4nn 8195 | . . . . . . 7 | |
11 | znq 8709 | . . . . . . 7 | |
12 | 10, 11 | mpan2 415 | . . . . . 6 |
13 | 12 | flqcld 9279 | . . . . 5 |
14 | zmulcl 8404 | . . . . 5 | |
15 | 2, 13, 14 | sylancr 405 | . . . 4 |
16 | ax-icn 7071 | . . . . 5 | |
17 | iap0 8254 | . . . . 5 # | |
18 | expsubap 9524 | . . . . 5 # | |
19 | 16, 17, 18 | mpanl12 426 | . . . 4 |
20 | 15, 19 | mpdan 412 | . . 3 |
21 | expmulzap 9522 | . . . . . . . 8 # | |
22 | 16, 17, 21 | mpanl12 426 | . . . . . . 7 |
23 | 2, 13, 22 | sylancr 405 | . . . . . 6 |
24 | i4 9577 | . . . . . . . 8 | |
25 | 24 | oveq1i 5542 | . . . . . . 7 |
26 | 1exp 9505 | . . . . . . . 8 | |
27 | 13, 26 | syl 14 | . . . . . . 7 |
28 | 25, 27 | syl5eq 2125 | . . . . . 6 |
29 | 23, 28 | eqtrd 2113 | . . . . 5 |
30 | 29 | oveq2d 5548 | . . . 4 |
31 | expclzap 9501 | . . . . . 6 # | |
32 | 16, 17, 31 | mp3an12 1258 | . . . . 5 |
33 | 32 | div1d 7868 | . . . 4 |
34 | 30, 33 | eqtrd 2113 | . . 3 |
35 | 20, 34 | eqtrd 2113 | . 2 |
36 | 9, 35 | eqtrd 2113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 class class class wbr 3785 cfv 4922 (class class class)co 5532 cc 6979 cc0 6981 c1 6982 ci 6983 cmul 6986 clt 7153 cmin 7279 # cap 7681 cdiv 7760 cn 8039 c4 8091 cz 8351 cq 8704 cfl 9272 cmo 9324 cexp 9475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-uz 8620 df-q 8705 df-rp 8735 df-fl 9274 df-mod 9325 df-iseq 9432 df-iexp 9476 |
This theorem is referenced by: (None) |
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