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Mirrors > Home > MPE Home > Th. List > dvdsmulc | Structured version Visualization version Unicode version |
Description: Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdsmulc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpc 1060 | . . 3 | |
2 | zmulcl 11426 | . . . . . 6 | |
3 | 2 | 3adant2 1080 | . . . . 5 |
4 | zmulcl 11426 | . . . . . 6 | |
5 | 4 | 3adant1 1079 | . . . . 5 |
6 | 3, 5 | jca 554 | . . . 4 |
7 | 6 | 3comr 1273 | . . 3 |
8 | simpr 477 | . . 3 | |
9 | zcn 11382 | . . . . . . . . 9 | |
10 | zcn 11382 | . . . . . . . . 9 | |
11 | zcn 11382 | . . . . . . . . 9 | |
12 | mulass 10024 | . . . . . . . . 9 | |
13 | 9, 10, 11, 12 | syl3an 1368 | . . . . . . . 8 |
14 | 13 | 3com13 1270 | . . . . . . 7 |
15 | 14 | 3expa 1265 | . . . . . 6 |
16 | 15 | 3adantl3 1219 | . . . . 5 |
17 | oveq1 6657 | . . . . 5 | |
18 | 16, 17 | sylan9req 2677 | . . . 4 |
19 | 18 | ex 450 | . . 3 |
20 | 1, 7, 8, 19 | dvds1lem 14993 | . 2 |
21 | 20 | 3coml 1272 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 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: dvdsmulcr 15011 coprmdvds2 15368 mulgcddvds 15369 rpmulgcd2 15370 pcpremul 15548 odadd2 18252 ablfacrp2 18466 znrrg 19914 dvdsmulf1o 24920 |
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