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Mirrors > Home > ILE Home > Th. List > rplpwr | Unicode version |
Description: If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
rplpwr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5844 | . . . . . . . 8 | |
2 | 1 | oveq1d 5851 | . . . . . . 7 |
3 | 2 | eqeq1d 2173 | . . . . . 6 |
4 | 3 | imbi2d 229 | . . . . 5 |
5 | oveq2 5844 | . . . . . . . 8 | |
6 | 5 | oveq1d 5851 | . . . . . . 7 |
7 | 6 | eqeq1d 2173 | . . . . . 6 |
8 | 7 | imbi2d 229 | . . . . 5 |
9 | oveq2 5844 | . . . . . . . 8 | |
10 | 9 | oveq1d 5851 | . . . . . . 7 |
11 | 10 | eqeq1d 2173 | . . . . . 6 |
12 | 11 | imbi2d 229 | . . . . 5 |
13 | oveq2 5844 | . . . . . . . 8 | |
14 | 13 | oveq1d 5851 | . . . . . . 7 |
15 | 14 | eqeq1d 2173 | . . . . . 6 |
16 | 15 | imbi2d 229 | . . . . 5 |
17 | nncn 8856 | . . . . . . . . . 10 | |
18 | 17 | exp1d 10572 | . . . . . . . . 9 |
19 | 18 | oveq1d 5851 | . . . . . . . 8 |
20 | 19 | adantr 274 | . . . . . . 7 |
21 | 20 | eqeq1d 2173 | . . . . . 6 |
22 | 21 | biimpar 295 | . . . . 5 |
23 | df-3an 969 | . . . . . . . . 9 | |
24 | simpl1 989 | . . . . . . . . . . . . . . . . 17 | |
25 | 24 | nncnd 8862 | . . . . . . . . . . . . . . . 16 |
26 | simpl3 991 | . . . . . . . . . . . . . . . . 17 | |
27 | 26 | nnnn0d 9158 | . . . . . . . . . . . . . . . 16 |
28 | 25, 27 | expp1d 10578 | . . . . . . . . . . . . . . 15 |
29 | simp1 986 | . . . . . . . . . . . . . . . . . . . 20 | |
30 | nnnn0 9112 | . . . . . . . . . . . . . . . . . . . . 21 | |
31 | 30 | 3ad2ant3 1009 | . . . . . . . . . . . . . . . . . . . 20 |
32 | 29, 31 | nnexpcld 10599 | . . . . . . . . . . . . . . . . . . 19 |
33 | 32 | nnzd 9303 | . . . . . . . . . . . . . . . . . 18 |
34 | 33 | adantr 274 | . . . . . . . . . . . . . . . . 17 |
35 | 34 | zcnd 9305 | . . . . . . . . . . . . . . . 16 |
36 | 35, 25 | mulcomd 7911 | . . . . . . . . . . . . . . 15 |
37 | 28, 36 | eqtrd 2197 | . . . . . . . . . . . . . 14 |
38 | 37 | oveq2d 5852 | . . . . . . . . . . . . 13 |
39 | simpl2 990 | . . . . . . . . . . . . . 14 | |
40 | 32 | adantr 274 | . . . . . . . . . . . . . 14 |
41 | nnz 9201 | . . . . . . . . . . . . . . . . . 18 | |
42 | 41 | 3ad2ant1 1007 | . . . . . . . . . . . . . . . . 17 |
43 | nnz 9201 | . . . . . . . . . . . . . . . . . 18 | |
44 | 43 | 3ad2ant2 1008 | . . . . . . . . . . . . . . . . 17 |
45 | gcdcom 11891 | . . . . . . . . . . . . . . . . 17 | |
46 | 42, 44, 45 | syl2anc 409 | . . . . . . . . . . . . . . . 16 |
47 | 46 | eqeq1d 2173 | . . . . . . . . . . . . . . 15 |
48 | 47 | biimpa 294 | . . . . . . . . . . . . . 14 |
49 | rpmulgcd 11944 | . . . . . . . . . . . . . 14 | |
50 | 39, 24, 40, 48, 49 | syl31anc 1230 | . . . . . . . . . . . . 13 |
51 | 38, 50 | eqtrd 2197 | . . . . . . . . . . . 12 |
52 | peano2nn 8860 | . . . . . . . . . . . . . . . . . 18 | |
53 | 52 | 3ad2ant3 1009 | . . . . . . . . . . . . . . . . 17 |
54 | 53 | adantr 274 | . . . . . . . . . . . . . . . 16 |
55 | 54 | nnnn0d 9158 | . . . . . . . . . . . . . . 15 |
56 | 24, 55 | nnexpcld 10599 | . . . . . . . . . . . . . 14 |
57 | 56 | nnzd 9303 | . . . . . . . . . . . . 13 |
58 | 44 | adantr 274 | . . . . . . . . . . . . 13 |
59 | gcdcom 11891 | . . . . . . . . . . . . 13 | |
60 | 57, 58, 59 | syl2anc 409 | . . . . . . . . . . . 12 |
61 | gcdcom 11891 | . . . . . . . . . . . . 13 | |
62 | 34, 58, 61 | syl2anc 409 | . . . . . . . . . . . 12 |
63 | 51, 60, 62 | 3eqtr4d 2207 | . . . . . . . . . . 11 |
64 | 63 | eqeq1d 2173 | . . . . . . . . . 10 |
65 | 64 | biimprd 157 | . . . . . . . . 9 |
66 | 23, 65 | sylanbr 283 | . . . . . . . 8 |
67 | 66 | an32s 558 | . . . . . . 7 |
68 | 67 | expcom 115 | . . . . . 6 |
69 | 68 | a2d 26 | . . . . 5 |
70 | 4, 8, 12, 16, 22, 69 | nnind 8864 | . . . 4 |
71 | 70 | expd 256 | . . 3 |
72 | 71 | com12 30 | . 2 |
73 | 72 | 3impia 1189 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 (class class class)co 5836 c1 7745 caddc 7747 cmul 7749 cn 8848 cn0 9105 cz 9182 cexp 10444 cgcd 11860 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-sup 6940 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-dvds 11714 df-gcd 11861 |
This theorem is referenced by: rppwr 11946 logbgcd1irr 13432 logbgcd1irraplemexp 13433 |
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