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Mirrors > Home > ILE Home > Th. List > dvdstr | 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 983 | . 2 | |
2 | 3simpc 985 | . 2 | |
3 | 3simpb 984 | . 2 | |
4 | zmulcl 9235 | . . 3 | |
5 | 4 | adantl 275 | . 2 |
6 | oveq2 5844 | . . . . 5 | |
7 | 6 | adantr 274 | . . . 4 |
8 | eqeq2 2174 | . . . . 5 | |
9 | 8 | adantl 275 | . . . 4 |
10 | 7, 9 | mpbid 146 | . . 3 |
11 | zcn 9187 | . . . . . . . 8 | |
12 | zcn 9187 | . . . . . . . 8 | |
13 | zcn 9187 | . . . . . . . 8 | |
14 | mulass 7875 | . . . . . . . . 9 | |
15 | mul12 8018 | . . . . . . . . 9 | |
16 | 14, 15 | eqtrd 2197 | . . . . . . . 8 |
17 | 11, 12, 13, 16 | syl3an 1269 | . . . . . . 7 |
18 | 17 | 3comr 1200 | . . . . . 6 |
19 | 18 | 3expb 1193 | . . . . 5 |
20 | 19 | 3ad2antl1 1148 | . . . 4 |
21 | 20 | eqeq1d 2173 | . . 3 |
22 | 10, 21 | syl5ibr 155 | . 2 |
23 | 1, 2, 3, 5, 22 | dvds2lem 11729 | 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 cc 7742 cmul 7749 cz 9182 cdvds 11713 |
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-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 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-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 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-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-dvds 11714 |
This theorem is referenced by: dvdstrd 11755 dvdsmultr1 11756 dvdsmultr2 11758 4dvdseven 11839 dvdsgcdb 11931 dvdsmulgcd 11943 gcddvdslcm 11984 lcmgcdeq 11994 lcmdvdsb 11995 mulgcddvds 12005 rpmulgcd2 12006 rpdvds 12010 exprmfct 12049 rpexp 12064 phimullem 12136 pcpremul 12204 pcdvdsb 12230 pcprmpw2 12243 |
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