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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 10578. (Contributed by Mario Carneiro, 7-Jul-2014.) |
Ref | Expression |
---|---|
iexpcyc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zq 9585 | . . . 4 | |
2 | 4z 9242 | . . . . . 6 | |
3 | zq 9585 | . . . . . 6 | |
4 | 2, 3 | ax-mp 5 | . . . . 5 |
5 | 4pos 8975 | . . . . 5 | |
6 | modqval 10280 | . . . . 5 | |
7 | 4, 5, 6 | mp3an23 1324 | . . . 4 |
8 | 1, 7 | syl 14 | . . 3 |
9 | 8 | oveq2d 5869 | . 2 |
10 | 4nn 9041 | . . . . . . 7 | |
11 | znq 9583 | . . . . . . 7 | |
12 | 10, 11 | mpan2 423 | . . . . . 6 |
13 | 12 | flqcld 10233 | . . . . 5 |
14 | zmulcl 9265 | . . . . 5 | |
15 | 2, 13, 14 | sylancr 412 | . . . 4 |
16 | ax-icn 7869 | . . . . 5 | |
17 | iap0 9101 | . . . . 5 # | |
18 | expsubap 10524 | . . . . 5 # | |
19 | 16, 17, 18 | mpanl12 434 | . . . 4 |
20 | 15, 19 | mpdan 419 | . . 3 |
21 | expmulzap 10522 | . . . . . . . 8 # | |
22 | 16, 17, 21 | mpanl12 434 | . . . . . . 7 |
23 | 2, 13, 22 | sylancr 412 | . . . . . 6 |
24 | i4 10578 | . . . . . . . 8 | |
25 | 24 | oveq1i 5863 | . . . . . . 7 |
26 | 1exp 10505 | . . . . . . . 8 | |
27 | 13, 26 | syl 14 | . . . . . . 7 |
28 | 25, 27 | eqtrid 2215 | . . . . . 6 |
29 | 23, 28 | eqtrd 2203 | . . . . 5 |
30 | 29 | oveq2d 5869 | . . . 4 |
31 | expclzap 10501 | . . . . . 6 # | |
32 | 16, 17, 31 | mp3an12 1322 | . . . . 5 |
33 | 32 | div1d 8697 | . . . 4 |
34 | 30, 33 | eqtrd 2203 | . . 3 |
35 | 20, 34 | eqtrd 2203 | . 2 |
36 | 9, 35 | eqtrd 2203 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 class class class wbr 3989 cfv 5198 (class class class)co 5853 cc 7772 cc0 7774 c1 7775 ci 7776 cmul 7779 clt 7954 cmin 8090 # cap 8500 cdiv 8589 cn 8878 c4 8931 cz 9212 cq 9578 cfl 10224 cmo 10278 cexp 10475 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 |
This theorem is referenced by: (None) |
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