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| Mirrors > Home > MPE Home > Th. List > odadd2 | Structured version Visualization version Unicode version | ||
| Description: The order of a product in an abelian group is divisible by the LCM of the orders of the factors divided by the GCD. (Contributed by Mario Carneiro, 20-Oct-2015.) |
| Ref | Expression |
|---|---|
| odadd1.1 |
        |
| odadd1.2 |
  |
| odadd1.3 |
  |
| Ref | Expression |
|---|---|
| odadd2 |
  ![/\](wedge.gif "/\"){kind=small} 
       |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | odadd1.2 |
. . . . . . . . 9
| |
| 2 | odadd1.1 |
. . . . . . . . 9
| |
| 3 | 1, 2 | odcl 17955 |
. . . . . . . 8
 |
| 4 | 3 | 3ad2ant2 1083 |
. . . . . . 7
|
| 5 | 4 | nn0zd 11480 |
. . . . . 6
 |
| 6 | 1, 2 | odcl 17955 |
. . . . . . . 8
|
| 7 | 6 | 3ad2ant3 1084 |
. . . . . . 7
|
| 8 | 7 | nn0zd 11480 |
. . . . . 6
|
| 9 | 5, 8 | zmulcld 11488 |
. . . . 5
|
| 10 | 9 | adantr 481 |
. . . 4
 |
| 11 | dvds0 14997 |
. . . 4
| |
| 12 | 10, 11 | syl 17 |
. . 3
|
| 13 | simpr 477 |
. . . . . 6
| |
| 14 | 13 | sq0id 12957 |
. . . . 5
|
| 15 | 14 | oveq2d 6666 |
. . . 4
|
| 16 | ablgrp 18198 |
. . . . . . . . . 10
 | |
| 17 | odadd1.3 |
. . . . . . . . . . 11
| |
| 18 | 1, 17 | grpcl 17430 |
. . . . . . . . . 10
|
| 19 | 16, 18 | syl3an1 1359 |
. . . . . . . . 9
|
| 20 | 1, 2 | odcl 17955 |
. . . . . . . . 9
|
| 21 | 19, 20 | syl 17 |
. . . . . . . 8
|
| 22 | 21 | nn0zd 11480 |
. . . . . . 7
|
| 23 | 22 | adantr 481 |
. . . . . 6
|
| 24 | 23 | zcnd 11483 |
. . . . 5
 |
| 25 | 24 | mul01d 10235 |
. . . 4
|
| 26 | 15, 25 | eqtrd 2656 |
. . 3
|
| 27 | 12, 26 | breqtrrd 4681 |
. 2
|
| 28 | 5 | adantr 481 |
. . . . . . . 8
 |
| 29 | 8 | adantr 481 |
. . . . . . . 8
|
| 30 | 28, 29 | gcdcld 15230 |
. . . . . . 7
|
| 31 | 30 | nn0cnd 11353 |
. . . . . 6
|
| 32 | 31 | sqvald 13005 |
. . . . 5
|
| 33 | 32 | oveq2d 6666 |
. . . 4
|
| 34 | gcddvds 15225 |
. . . . . . . . 9
| |
| 35 | 28, 29, 34 | syl2anc 693 |
. . . . . . . 8
|
| 36 | 35 | simpld 475 |
. . . . . . 7
|
| 37 | 30 | nn0zd 11480 |
. . . . . . . 8
|
| 38 | simpr 477 |
. . . . . . . 8
| |
| 39 | dvdsval2 14986 |
. . . . . . . 8
| |
| 40 | 37, 38, 28, 39 | syl3anc 1326 |
. . . . . . 7
|
| 41 | 36, 40 | mpbid 222 |
. . . . . 6
|
| 42 | 41 | zcnd 11483 |
. . . . 5
|
| 43 | 35 | simprd 479 |
. . . . . . 7
|
| 44 | dvdsval2 14986 |
. . . . . . . 8
| |
| 45 | 37, 38, 29, 44 | syl3anc 1326 |
. . . . . . 7
|
| 46 | 43, 45 | mpbid 222 |
. . . . . 6
|
| 47 | 46 | zcnd 11483 |
. . . . 5
|
| 48 | 42, 31, 47, 31 | mul4d 10248 |
. . . 4
|
| 49 | 28 | zcnd 11483 |
. . . . . 6
|
| 50 | 49, 31, 38 | divcan1d 10802 |
. . . . 5
|
| 51 | 29 | zcnd 11483 |
. . . . . 6
|
| 52 | 51, 31, 38 | divcan1d 10802 |
. . . . 5
|
| 53 | 50, 52 | oveq12d 6668 |
. . . 4
|
| 54 | 33, 48, 53 | 3eqtr2d 2662 |
. . 3
|
| 55 | 22 | adantr 481 |
. . . . . . . . . 10
|
| 56 | dvdsmul2 15004 |
. . . . . . . . . 10
| |
| 57 | 55, 28, 56 | syl2anc 693 |
. . . . . . . . 9
|
| 58 | simpl1 1064 |
. . . . . . . . . . . . 13
| |
| 59 | 55, 29 | zmulcld 11488 |
. . . . . . . . . . . . 13
|
| 60 | simpl2 1065 |
. . . . . . . . . . . . 13
| |
| 61 | simpl3 1066 |
. . . . . . . . . . . . 13
| |
| 62 | eqid 2622 |
. . . . . . . . . . . . . 14
.g .g | |
| 63 | 1, 62, 17 | mulgdi 18232 |
. . . . . . . . . . . . 13
.g .g .g |
| 64 | 58, 59, 60, 61, 63 | syl13anc 1328 |
. . . . . . . . . . . 12
.g .g .g |
| 65 | dvdsmul2 15004 |
. . . . . . . . . . . . . . 15
| |
| 66 | 55, 29, 65 | syl2anc 693 |
. . . . . . . . . . . . . 14
|
| 67 | 58, 16 | syl 17 |
. . . . . . . . . . . . . . 15
|
| 68 | eqid 2622 |
. . . . . . . . . . . . . . . 16
 | |
| 69 | 1, 2, 62, 68 | oddvds 17966 |
. . . . . . . . . . . . . . 15
.g |
| 70 | 67, 61, 59, 69 | syl3anc 1326 |
. . . . . . . . . . . . . 14
.g |
| 71 | 66, 70 | mpbid 222 |
. . . . . . . . . . . . 13
.g |
| 72 | 71 | oveq2d 6666 |
. . . . . . . . . . . 12
.g .g
.g |
| 73 | 64, 72 | eqtrd 2656 |
. . . . . . . . . . 11
.g .g |
| 74 | dvdsmul1 15003 |
. . . . . . . . . . . . 13
| |
| 75 | 55, 29, 74 | syl2anc 693 |
. . . . . . . . . . . 12
|
| 76 | 19 | adantr 481 |
. . . . . . . . . . . . 13
|
| 77 | 1, 2, 62, 68 | oddvds 17966 |
. . . . . . . . . . . . 13
.g |
| 78 | 67, 76, 59, 77 | syl3anc 1326 |
. . . . . . . . . . . 12
.g |
| 79 | 75, 78 | mpbid 222 |
. . . . . . . . . . 11
.g |
| 80 | 1, 62 | mulgcl 17559 |
. . . . . . . . . . . . 13
.g |
| 81 | 67, 59, 60, 80 | syl3anc 1326 |
. . . . . . . . . . . 12
.g |
| 82 | 1, 17, 68 | grprid 17453 |
. . . . . . . . . . . 12
.g .g .g |
| 83 | 67, 81, 82 | syl2anc 693 |
. . . . . . . . . . 11
.g .g |
| 84 | 73, 79, 83 | 3eqtr3rd 2665 |
. . . . . . . . . 10
.g |
| 85 | 1, 2, 62, 68 | oddvds 17966 |
. . . . . . . . . . 11
.g |
| 86 | 67, 60, 59, 85 | syl3anc 1326 |
. . . . . . . . . 10
.g |
| 87 | 84, 86 | mpbird 247 |
. . . . . . . . 9
|
| 88 | 55, 28 | zmulcld 11488 |
. . . . . . . . . 10
|
| 89 | dvdsgcd 15261 |
. . . . . . . . . 10
| |
| 90 | 28, 88, 59, 89 | syl3anc 1326 |
. . . . . . . . 9
|
| 91 | 57, 87, 90 | mp2and 715 |
. . . . . . . 8
|
| 92 | 21 | adantr 481 |
. . . . . . . . 9
|
| 93 | mulgcd 15265 |
. . . . . . . . 9
| |
| 94 | 92, 28, 29, 93 | syl3anc 1326 |
. . . . . . . 8
|
| 95 | 91, 94 | breqtrd 4679 |
. . . . . . 7
|
| 96 | 50, 95 | eqbrtrd 4675 |
. . . . . 6
|
| 97 | dvdsmulcr 15011 |
. . . . . . 7
| |
| 98 | 41, 55, 37, 38, 97 | syl112anc 1330 |
. . . . . 6
|
| 99 | 96, 98 | mpbid 222 |
. . . . 5
|
| 100 | 1, 62, 17 | mulgdi 18232 |
. . . . . . . . . . . . 13
.g .g .g |
| 101 | 58, 88, 60, 61, 100 | syl13anc 1328 |
. . . . . . . . . . . 12
.g .g .g |
| 102 | 1, 2, 62, 68 | oddvds 17966 |
. . . . . . . . . . . . . . 15
.g |
| 103 | 67, 60, 88, 102 | syl3anc 1326 |
. . . . . . . . . . . . . 14
.g |
| 104 | 57, 103 | mpbid 222 |
. . . . . . . . . . . . 13
.g |
| 105 | 104 | oveq1d 6665 |
. . . . . . . . . . . 12
.g .g .g |
| 106 | 101, 105 | eqtrd 2656 |
. . . . . . . . . . 11
.g .g |
| 107 | dvdsmul1 15003 |
. . . . . . . . . . . . 13
| |
| 108 | 55, 28, 107 | syl2anc 693 |
. . . . . . . . . . . 12
|
| 109 | 1, 2, 62, 68 | oddvds 17966 |
. . . . . . . . . . . . 13
.g |
| 110 | 67, 76, 88, 109 | syl3anc 1326 |
. . . . . . . . . . . 12
.g |
| 111 | 108, 110 | mpbid 222 |
. . . . . . . . . . 11
.g |
| 112 | 1, 62 | mulgcl 17559 |
. . . . . . . . . . . . 13
.g |
| 113 | 67, 88, 61, 112 | syl3anc 1326 |
. . . . . . . . . . . 12
.g |
| 114 | 1, 17, 68 | grplid 17452 |
. . . . . . . . . . . 12
.g
.g .g |
| 115 | 67, 113, 114 | syl2anc 693 |
. . . . . . . . . . 11
.g .g |
| 116 | 106, 111, 115 | 3eqtr3rd 2665 |
. . . . . . . . . 10
.g |
| 117 | 1, 2, 62, 68 | oddvds 17966 |
. . . . . . . . . . 11
.g |
| 118 | 67, 61, 88, 117 | syl3anc 1326 |
. . . . . . . . . 10
.g |
| 119 | 116, 118 | mpbird 247 |
. . . . . . . . 9
|
| 120 | dvdsgcd 15261 |
. . . . . . . . . 10
| |
| 121 | 29, 88, 59, 120 | syl3anc 1326 |
. . . . . . . . 9
|
| 122 | 119, 66, 121 | mp2and 715 |
. . . . . . . 8
|
| 123 | 122, 94 | breqtrd 4679 |
. . . . . . 7
|
| 124 | 52, 123 | eqbrtrd 4675 |
. . . . . 6
|
| 125 | dvdsmulcr 15011 |
. . . . . . 7
| |
| 126 | 46, 55, 37, 38, 125 | syl112anc 1330 |
. . . . . 6
|
| 127 | 124, 126 | mpbid 222 |
. . . . 5
|
| 128 | 41, 46 | gcdcld 15230 |
. . . . . . . 8
|
| 129 | 128 | nn0cnd 11353 |
. . . . . . 7
|
| 130 | 1cnd 10056 |
. . . . . . 7
 | |
| 131 | 31 | mulid2d 10058 |
. . . . . . . 8
|
| 132 | 50, 52 | oveq12d 6668 |
. . . . . . . 8
|
| 133 | mulgcdr 15267 |
. . . . . . . . 9
| |
| 134 | 41, 46, 30, 133 | syl3anc 1326 |
. . . . . . . 8
|
| 135 | 131, 132, 134 | 3eqtr2rd 2663 |
. . . . . . 7
|
| 136 | 129, 130, 31, 38, 135 | mulcan2ad 10663 |
. . . . . 6
|
| 137 | coprmdvds2 15368 |
. . . . . 6
| |
| 138 | 41, 46, 55, 136, 137 | syl31anc 1329 |
. . . . 5
|
| 139 | 99, 127, 138 | mp2and 715 |
. . . 4
|
| 140 | 41, 46 | zmulcld 11488 |
. . . . 5
|
| 141 | zsqcl 12934 |
. . . . . 6
| |
| 142 | 37, 141 | syl 17 |
. . . . 5
|
| 143 | dvdsmulc 15009 |
. . . . 5
| |
| 144 | 140, 55, 142, 143 | syl3anc 1326 |
. . . 4
|
| 145 | 139, 144 | mpd 15 |
. . 3
|
| 146 | 54, 145 | eqbrtrrd 4677 |
. 2
|
| 147 | 27, 146 | pm2.61dane 2881 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wi 4
wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 class class class wbr 4653
cfv 5888
(class class class)co 6650 cc0 9936 c1 9937 cmul 9941 cdiv 10684 c2 11070 cn0 11292 cz 11377 cexp 12860 cdvds 14983 cgcd 15216 cbs 15857 cplusg 15941 c0g 16100 cgrp 17422 .gcmg 17540
cod 17944
cabl 18194 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-od 17948 df-cmn 18195 df-abl 18196 |
| This theorem is referenced by: odadd 18253 |
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