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Mirrors > Home > ILE Home > Th. List > dvds2ln | Unicode version |
Description: If an integer divides each of two other integers, it divides any linear combination of them. Theorem 1.1(c) in [ApostolNT] p. 14 (linearity property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvds2ln |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 987 | . . 3 | |
2 | simpr2 988 | . . 3 | |
3 | 1, 2 | jca 304 | . 2 |
4 | simpr3 989 | . . 3 | |
5 | 1, 4 | jca 304 | . 2 |
6 | simpll 518 | . . . . 5 | |
7 | 6, 2 | zmulcld 9179 | . . . 4 |
8 | simplr 519 | . . . . 5 | |
9 | 8, 4 | zmulcld 9179 | . . . 4 |
10 | 7, 9 | zaddcld 9177 | . . 3 |
11 | 1, 10 | jca 304 | . 2 |
12 | zmulcl 9107 | . . . . . . . 8 | |
13 | zmulcl 9107 | . . . . . . . 8 | |
14 | 12, 13 | anim12i 336 | . . . . . . 7 |
15 | 14 | an4s 577 | . . . . . 6 |
16 | 15 | expcom 115 | . . . . 5 |
17 | 16 | adantr 274 | . . . 4 |
18 | 17 | imp 123 | . . 3 |
19 | zaddcl 9094 | . . 3 | |
20 | 18, 19 | syl 14 | . 2 |
21 | zcn 9059 | . . . . . . . 8 | |
22 | zcn 9059 | . . . . . . . 8 | |
23 | 21, 22 | anim12i 336 | . . . . . . 7 |
24 | 18, 23 | syl 14 | . . . . . 6 |
25 | 1 | zcnd 9174 | . . . . . . 7 |
26 | 25 | adantr 274 | . . . . . 6 |
27 | adddir 7757 | . . . . . . 7 | |
28 | 27 | 3expa 1181 | . . . . . 6 |
29 | 24, 26, 28 | syl2anc 408 | . . . . 5 |
30 | zcn 9059 | . . . . . . . . 9 | |
31 | 30 | adantr 274 | . . . . . . . 8 |
32 | 31 | adantl 275 | . . . . . . 7 |
33 | zcn 9059 | . . . . . . . 8 | |
34 | 33 | ad3antrrr 483 | . . . . . . 7 |
35 | 32, 34, 26 | mul32d 7915 | . . . . . 6 |
36 | zcn 9059 | . . . . . . . . 9 | |
37 | 36 | adantl 275 | . . . . . . . 8 |
38 | 37 | adantl 275 | . . . . . . 7 |
39 | 8 | zcnd 9174 | . . . . . . . 8 |
40 | 39 | adantr 274 | . . . . . . 7 |
41 | 38, 40, 26 | mul32d 7915 | . . . . . 6 |
42 | 35, 41 | oveq12d 5792 | . . . . 5 |
43 | 32, 26 | mulcld 7786 | . . . . . . 7 |
44 | 43, 34 | mulcomd 7787 | . . . . . 6 |
45 | 38, 26 | mulcld 7786 | . . . . . . 7 |
46 | 45, 40 | mulcomd 7787 | . . . . . 6 |
47 | 44, 46 | oveq12d 5792 | . . . . 5 |
48 | 29, 42, 47 | 3eqtrd 2176 | . . . 4 |
49 | oveq2 5782 | . . . . 5 | |
50 | oveq2 5782 | . . . . 5 | |
51 | 49, 50 | oveqan12d 5793 | . . . 4 |
52 | 48, 51 | sylan9eq 2192 | . . 3 |
53 | 52 | ex 114 | . 2 |
54 | 3, 5, 11, 20, 53 | dvds2lem 11505 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 class class class wbr 3929 (class class class)co 5774 cc 7618 caddc 7623 cmul 7625 cz 9054 cdvds 11493 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-dvds 11494 |
This theorem is referenced by: gcdaddm 11672 dvdsgcd 11700 |
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