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Mirrors > Home > MPE Home > Th. List > dvdstr | Structured version Visualization version Unicode version |
Description: The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14 (transitive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdstr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 1058 | . 2 | |
2 | 3simpc 1060 | . 2 | |
3 | 3simpb 1059 | . 2 | |
4 | zmulcl 11426 | . . 3 | |
5 | 4 | adantl 482 | . 2 |
6 | oveq2 6658 | . . . . 5 | |
7 | 6 | adantr 481 | . . . 4 |
8 | eqeq2 2633 | . . . . 5 | |
9 | 8 | adantl 482 | . . . 4 |
10 | 7, 9 | mpbid 222 | . . 3 |
11 | zcn 11382 | . . . . . . . 8 | |
12 | zcn 11382 | . . . . . . . 8 | |
13 | zcn 11382 | . . . . . . . 8 | |
14 | mulass 10024 | . . . . . . . . 9 | |
15 | mul12 10202 | . . . . . . . . 9 | |
16 | 14, 15 | eqtrd 2656 | . . . . . . . 8 |
17 | 11, 12, 13, 16 | syl3an 1368 | . . . . . . 7 |
18 | 17 | 3comr 1273 | . . . . . 6 |
19 | 18 | 3expb 1266 | . . . . 5 |
20 | 19 | 3ad2antl1 1223 | . . . 4 |
21 | 20 | eqeq1d 2624 | . . 3 |
22 | 10, 21 | syl5ibr 236 | . 2 |
23 | 1, 2, 3, 5, 22 | dvds2lem 14994 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 class class class wbr 4653 (class class class)co 6650 cc 9934 cmul 9941 cz 11377 cdvds 14983 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 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 |
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-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-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-dvds 14984 |
This theorem is referenced by: dvdsmultr1 15019 dvdsmultr2 15021 4dvdseven 15109 bitsmod 15158 dvdsgcdb 15262 dvdsmulgcd 15274 gcddvdslcm 15315 lcmgcdeq 15325 lcmdvdsb 15326 lcmftp 15349 lcmfunsnlem2lem2 15352 lcmfdvdsb 15356 mulgcddvds 15369 rpmulgcd2 15370 rpdvds 15374 exprmfct 15416 isprm5 15419 rpexp 15432 phimullem 15484 pcpremul 15548 pcdvdsb 15573 pcdvdstr 15580 pcprmpw2 15586 pockthlem 15609 prmreclem3 15622 4sqlem8 15649 odmulg 17973 ablfac1b 18469 ablfac1eu 18472 znunit 19912 wilth 24797 muval1 24859 dvdssqf 24864 sqff1o 24908 fsumdvdsdiaglem 24909 dvdsmulf1o 24920 vmasum 24941 bposlem3 25011 lgsmod 25048 lgsquad2lem1 25109 2sqlem3 25145 2sqlem8 25151 dvdspw 31636 dvdsacongtr 37551 jm2.20nn 37564 jm2.27a 37572 jm2.27c 37574 goldbachthlem2 41458 |
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