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Mirrors > Home > ILE Home > Th. List > rpmulgcd2 | Unicode version |
Description: If is relatively prime to , then the GCD of with is the product of the GCDs with and respectively. (Contributed by Mario Carneiro, 2-Jul-2015.) |
Ref | Expression |
---|---|
rpmulgcd2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 989 | . . 3 | |
2 | simpl2 990 | . . . 4 | |
3 | simpl3 991 | . . . 4 | |
4 | 2, 3 | zmulcld 9310 | . . 3 |
5 | 1, 4 | gcdcld 11886 | . 2 |
6 | 1, 2 | gcdcld 11886 | . . 3 |
7 | 1, 3 | gcdcld 11886 | . . 3 |
8 | 6, 7 | nn0mulcld 9163 | . 2 |
9 | mulgcddvds 12005 | . . 3 | |
10 | 9 | adantr 274 | . 2 |
11 | gcddvds 11881 | . . . . . 6 | |
12 | 1, 2, 11 | syl2anc 409 | . . . . 5 |
13 | 12 | simpld 111 | . . . 4 |
14 | gcddvds 11881 | . . . . . 6 | |
15 | 1, 3, 14 | syl2anc 409 | . . . . 5 |
16 | 15 | simpld 111 | . . . 4 |
17 | 6 | nn0zd 9302 | . . . . 5 |
18 | 7 | nn0zd 9302 | . . . . 5 |
19 | gcddvds 11881 | . . . . . . . . . . 11 | |
20 | 17, 18, 19 | syl2anc 409 | . . . . . . . . . 10 |
21 | 20 | simpld 111 | . . . . . . . . 9 |
22 | 12 | simprd 113 | . . . . . . . . 9 |
23 | 17, 18 | gcdcld 11886 | . . . . . . . . . . 11 |
24 | 23 | nn0zd 9302 | . . . . . . . . . 10 |
25 | dvdstr 11754 | . . . . . . . . . 10 | |
26 | 24, 17, 2, 25 | syl3anc 1227 | . . . . . . . . 9 |
27 | 21, 22, 26 | mp2and 430 | . . . . . . . 8 |
28 | 20 | simprd 113 | . . . . . . . . 9 |
29 | 15 | simprd 113 | . . . . . . . . 9 |
30 | dvdstr 11754 | . . . . . . . . . 10 | |
31 | 24, 18, 3, 30 | syl3anc 1227 | . . . . . . . . 9 |
32 | 28, 29, 31 | mp2and 430 | . . . . . . . 8 |
33 | dvdsgcd 11930 | . . . . . . . . 9 | |
34 | 24, 2, 3, 33 | syl3anc 1227 | . . . . . . . 8 |
35 | 27, 32, 34 | mp2and 430 | . . . . . . 7 |
36 | simpr 109 | . . . . . . 7 | |
37 | 35, 36 | breqtrd 4002 | . . . . . 6 |
38 | dvds1 11776 | . . . . . . 7 | |
39 | 23, 38 | syl 14 | . . . . . 6 |
40 | 37, 39 | mpbid 146 | . . . . 5 |
41 | coprmdvds2 12004 | . . . . 5 | |
42 | 17, 18, 1, 40, 41 | syl31anc 1230 | . . . 4 |
43 | 13, 16, 42 | mp2and 430 | . . 3 |
44 | dvdscmul 11744 | . . . . . 6 | |
45 | 18, 3, 17, 44 | syl3anc 1227 | . . . . 5 |
46 | dvdsmulc 11745 | . . . . . 6 | |
47 | 17, 2, 3, 46 | syl3anc 1227 | . . . . 5 |
48 | 17, 18 | zmulcld 9310 | . . . . . 6 |
49 | 17, 3 | zmulcld 9310 | . . . . . 6 |
50 | dvdstr 11754 | . . . . . 6 | |
51 | 48, 49, 4, 50 | syl3anc 1227 | . . . . 5 |
52 | 45, 47, 51 | syl2and 293 | . . . 4 |
53 | 29, 22, 52 | mp2and 430 | . . 3 |
54 | dvdsgcd 11930 | . . . 4 | |
55 | 48, 1, 4, 54 | syl3anc 1227 | . . 3 |
56 | 43, 53, 55 | mp2and 430 | . 2 |
57 | dvdseq 11771 | . 2 | |
58 | 5, 8, 10, 56, 57 | syl22anc 1228 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 class class class wbr 3976 (class class class)co 5836 c1 7745 cmul 7749 cn0 9105 cz 9182 cdvds 11713 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-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: (None) |
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