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Mirrors > Home > MPE Home > Th. List > dvdsmulgcd | Structured version Visualization version Unicode version |
Description: A divisibility equivalent for odmulg 17973. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
dvdsmulgcd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 792 | . . . 4 | |
2 | dvdszrcl 14988 | . . . . . 6 | |
3 | 2 | adantl 482 | . . . . 5 |
4 | 3 | simpld 475 | . . . 4 |
5 | bezout 15260 | . . . 4 | |
6 | 1, 4, 5 | syl2anc 693 | . . 3 |
7 | 4 | adantr 481 | . . . . . . 7 |
8 | simplll 798 | . . . . . . . 8 | |
9 | simpllr 799 | . . . . . . . . 9 | |
10 | simprl 794 | . . . . . . . . 9 | |
11 | 9, 10 | zmulcld 11488 | . . . . . . . 8 |
12 | 8, 11 | zmulcld 11488 | . . . . . . 7 |
13 | simprr 796 | . . . . . . . . 9 | |
14 | 7, 13 | zmulcld 11488 | . . . . . . . 8 |
15 | 8, 14 | zmulcld 11488 | . . . . . . 7 |
16 | simplr 792 | . . . . . . . . 9 | |
17 | 8, 9 | zmulcld 11488 | . . . . . . . . . 10 |
18 | dvdsmultr1 15019 | . . . . . . . . . 10 | |
19 | 7, 17, 10, 18 | syl3anc 1326 | . . . . . . . . 9 |
20 | 16, 19 | mpd 15 | . . . . . . . 8 |
21 | 8 | zcnd 11483 | . . . . . . . . 9 |
22 | 9 | zcnd 11483 | . . . . . . . . 9 |
23 | 10 | zcnd 11483 | . . . . . . . . 9 |
24 | 21, 22, 23 | mulassd 10063 | . . . . . . . 8 |
25 | 20, 24 | breqtrd 4679 | . . . . . . 7 |
26 | 8, 13 | zmulcld 11488 | . . . . . . . . 9 |
27 | dvdsmul1 15003 | . . . . . . . . 9 | |
28 | 7, 26, 27 | syl2anc 693 | . . . . . . . 8 |
29 | 7 | zcnd 11483 | . . . . . . . . 9 |
30 | 13 | zcnd 11483 | . . . . . . . . 9 |
31 | 21, 29, 30 | mul12d 10245 | . . . . . . . 8 |
32 | 28, 31 | breqtrrd 4681 | . . . . . . 7 |
33 | dvds2add 15015 | . . . . . . . 8 | |
34 | 33 | imp 445 | . . . . . . 7 |
35 | 7, 12, 15, 25, 32, 34 | syl32anc 1334 | . . . . . 6 |
36 | 11 | zcnd 11483 | . . . . . . 7 |
37 | 14 | zcnd 11483 | . . . . . . 7 |
38 | 21, 36, 37 | adddid 10064 | . . . . . 6 |
39 | 35, 38 | breqtrrd 4681 | . . . . 5 |
40 | oveq2 6658 | . . . . . 6 | |
41 | 40 | breq2d 4665 | . . . . 5 |
42 | 39, 41 | syl5ibrcom 237 | . . . 4 |
43 | 42 | rexlimdvva 3038 | . . 3 |
44 | 6, 43 | mpd 15 | . 2 |
45 | dvdszrcl 14988 | . . . . 5 | |
46 | 45 | adantl 482 | . . . 4 |
47 | 46 | simpld 475 | . . 3 |
48 | 46 | simprd 479 | . . 3 |
49 | zmulcl 11426 | . . . 4 | |
50 | 49 | adantr 481 | . . 3 |
51 | simpr 477 | . . 3 | |
52 | simplr 792 | . . . . . 6 | |
53 | gcddvds 15225 | . . . . . 6 | |
54 | 52, 47, 53 | syl2anc 693 | . . . . 5 |
55 | 54 | simpld 475 | . . . 4 |
56 | 52, 47 | gcdcld 15230 | . . . . . 6 |
57 | 56 | nn0zd 11480 | . . . . 5 |
58 | simpll 790 | . . . . 5 | |
59 | dvdscmul 15008 | . . . . 5 | |
60 | 57, 52, 58, 59 | syl3anc 1326 | . . . 4 |
61 | 55, 60 | mpd 15 | . . 3 |
62 | dvdstr 15018 | . . . 4 | |
63 | 62 | imp 445 | . . 3 |
64 | 47, 48, 50, 51, 61, 63 | syl32anc 1334 | . 2 |
65 | 44, 64 | impbida 877 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wrex 2913 class class class wbr 4653 (class class class)co 6650 caddc 9939 cmul 9941 cz 11377 cdvds 14983 cgcd 15216 |
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-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-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-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 |
This theorem is referenced by: coprmdvds 15366 odmulg 17973 |
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