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Mirrors > Home > ILE Home > Th. List > dvdsmulgcd | Unicode version |
Description: Relationship between the order of an element and that of a multiple. (a divisibility equivalent). (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
dvdsmulgcd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 520 | . . . 4 | |
2 | dvdszrcl 11534 | . . . . . 6 | |
3 | 2 | adantl 275 | . . . . 5 |
4 | 3 | simpld 111 | . . . 4 |
5 | bezout 11735 | . . . 4 | |
6 | 1, 4, 5 | syl2anc 409 | . . 3 |
7 | 4 | adantr 274 | . . . . . . 7 |
8 | simplll 523 | . . . . . . . 8 | |
9 | simpllr 524 | . . . . . . . . 9 | |
10 | simprl 521 | . . . . . . . . 9 | |
11 | 9, 10 | zmulcld 9203 | . . . . . . . 8 |
12 | 8, 11 | zmulcld 9203 | . . . . . . 7 |
13 | simprr 522 | . . . . . . . . 9 | |
14 | 7, 13 | zmulcld 9203 | . . . . . . . 8 |
15 | 8, 14 | zmulcld 9203 | . . . . . . 7 |
16 | simplr 520 | . . . . . . . . 9 | |
17 | 8, 9 | zmulcld 9203 | . . . . . . . . . 10 |
18 | dvdsmultr1 11567 | . . . . . . . . . 10 | |
19 | 7, 17, 10, 18 | syl3anc 1217 | . . . . . . . . 9 |
20 | 16, 19 | mpd 13 | . . . . . . . 8 |
21 | 8 | zcnd 9198 | . . . . . . . . 9 |
22 | 9 | zcnd 9198 | . . . . . . . . 9 |
23 | 10 | zcnd 9198 | . . . . . . . . 9 |
24 | 21, 22, 23 | mulassd 7813 | . . . . . . . 8 |
25 | 20, 24 | breqtrd 3962 | . . . . . . 7 |
26 | 8, 13 | zmulcld 9203 | . . . . . . . . 9 |
27 | dvdsmul1 11551 | . . . . . . . . 9 | |
28 | 7, 26, 27 | syl2anc 409 | . . . . . . . 8 |
29 | 7 | zcnd 9198 | . . . . . . . . 9 |
30 | 13 | zcnd 9198 | . . . . . . . . 9 |
31 | 21, 29, 30 | mul12d 7938 | . . . . . . . 8 |
32 | 28, 31 | breqtrrd 3964 | . . . . . . 7 |
33 | dvds2add 11563 | . . . . . . . 8 | |
34 | 33 | imp 123 | . . . . . . 7 |
35 | 7, 12, 15, 25, 32, 34 | syl32anc 1225 | . . . . . 6 |
36 | 11 | zcnd 9198 | . . . . . . 7 |
37 | 14 | zcnd 9198 | . . . . . . 7 |
38 | 21, 36, 37 | adddid 7814 | . . . . . 6 |
39 | 35, 38 | breqtrrd 3964 | . . . . 5 |
40 | oveq2 5790 | . . . . . 6 | |
41 | 40 | breq2d 3949 | . . . . 5 |
42 | 39, 41 | syl5ibrcom 156 | . . . 4 |
43 | 42 | rexlimdvva 2560 | . . 3 |
44 | 6, 43 | mpd 13 | . 2 |
45 | dvdszrcl 11534 | . . . . 5 | |
46 | 45 | adantl 275 | . . . 4 |
47 | 46 | simpld 111 | . . 3 |
48 | 46 | simprd 113 | . . 3 |
49 | zmulcl 9131 | . . . 4 | |
50 | 49 | adantr 274 | . . 3 |
51 | simpr 109 | . . 3 | |
52 | simplr 520 | . . . . . 6 | |
53 | gcddvds 11688 | . . . . . 6 | |
54 | 52, 47, 53 | syl2anc 409 | . . . . 5 |
55 | 54 | simpld 111 | . . . 4 |
56 | 52, 47 | gcdcld 11693 | . . . . . 6 |
57 | 56 | nn0zd 9195 | . . . . 5 |
58 | simpll 519 | . . . . 5 | |
59 | dvdscmul 11556 | . . . . 5 | |
60 | 57, 52, 58, 59 | syl3anc 1217 | . . . 4 |
61 | 55, 60 | mpd 13 | . . 3 |
62 | dvdstr 11566 | . . . 4 | |
63 | 62 | imp 123 | . . 3 |
64 | 47, 48, 50, 51, 61, 63 | syl32anc 1225 | . 2 |
65 | 44, 64 | impbida 586 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1332 wcel 1481 wrex 2418 class class class wbr 3937 (class class class)co 5782 caddc 7647 cmul 7649 cz 9078 cdvds 11529 cgcd 11671 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-sup 6879 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-fz 9822 df-fzo 9951 df-fl 10074 df-mod 10127 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-dvds 11530 df-gcd 11672 |
This theorem is referenced by: coprmdvds 11809 |
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