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Mirrors > Home > ILE Home > Th. List > dvdsval2 | Unicode version |
Description: One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.) |
Ref | Expression |
---|---|
dvdsval2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divides 11495 | . . 3 | |
2 | 1 | 3adant2 1000 | . 2 |
3 | zcn 9059 | . . . . . . . . . . 11 | |
4 | 3 | 3ad2ant3 1004 | . . . . . . . . . 10 |
5 | 4 | adantr 274 | . . . . . . . . 9 |
6 | zcn 9059 | . . . . . . . . . 10 | |
7 | 6 | adantl 275 | . . . . . . . . 9 |
8 | zcn 9059 | . . . . . . . . . . 11 | |
9 | 8 | 3ad2ant1 1002 | . . . . . . . . . 10 |
10 | 9 | adantr 274 | . . . . . . . . 9 |
11 | simpl2 985 | . . . . . . . . . 10 | |
12 | 0z 9065 | . . . . . . . . . . . . 13 | |
13 | zapne 9125 | . . . . . . . . . . . . 13 # | |
14 | 12, 13 | mpan2 421 | . . . . . . . . . . . 12 # |
15 | 14 | 3ad2ant1 1002 | . . . . . . . . . . 11 # |
16 | 15 | adantr 274 | . . . . . . . . . 10 # |
17 | 11, 16 | mpbird 166 | . . . . . . . . 9 # |
18 | 5, 7, 10, 17 | divmulap3d 8585 | . . . . . . . 8 |
19 | eqcom 2141 | . . . . . . . 8 | |
20 | 18, 19 | syl6bb 195 | . . . . . . 7 |
21 | 20 | biimprd 157 | . . . . . 6 |
22 | 21 | impr 376 | . . . . 5 |
23 | simprl 520 | . . . . 5 | |
24 | 22, 23 | eqeltrd 2216 | . . . 4 |
25 | 24 | rexlimdvaa 2550 | . . 3 |
26 | simpr 109 | . . . . 5 | |
27 | simp2 982 | . . . . . . . 8 | |
28 | 27, 15 | mpbird 166 | . . . . . . 7 # |
29 | 4, 9, 28 | divcanap1d 8551 | . . . . . 6 |
30 | 29 | adantr 274 | . . . . 5 |
31 | oveq1 5781 | . . . . . . 7 | |
32 | 31 | eqeq1d 2148 | . . . . . 6 |
33 | 32 | rspcev 2789 | . . . . 5 |
34 | 26, 30, 33 | syl2anc 408 | . . . 4 |
35 | 34 | ex 114 | . . 3 |
36 | 25, 35 | impbid 128 | . 2 |
37 | 2, 36 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wne 2308 wrex 2417 class class class wbr 3929 (class class class)co 5774 cc 7618 cc0 7620 cmul 7625 # cap 8343 cdiv 8432 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-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
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-rmo 2424 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-po 4218 df-iso 4219 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-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-dvds 11494 |
This theorem is referenced by: dvdsval3 11497 nndivdvds 11499 divconjdvds 11547 zeo3 11565 evend2 11586 oddp1d2 11587 fldivndvdslt 11632 divgcdz 11660 dvdsgcdidd 11682 mulgcd 11704 sqgcd 11717 lcmgcdlem 11758 mulgcddvds 11775 qredeu 11778 prmind2 11801 divgcdodd 11821 divnumden 11874 hashdvds 11897 hashgcdlem 11903 |
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