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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 11685 | . . 3 | |
2 | 1 | 3adant2 1001 | . 2 |
3 | zcn 9172 | . . . . . . . . . . 11 | |
4 | 3 | 3ad2ant3 1005 | . . . . . . . . . 10 |
5 | 4 | adantr 274 | . . . . . . . . 9 |
6 | zcn 9172 | . . . . . . . . . 10 | |
7 | 6 | adantl 275 | . . . . . . . . 9 |
8 | zcn 9172 | . . . . . . . . . . 11 | |
9 | 8 | 3ad2ant1 1003 | . . . . . . . . . 10 |
10 | 9 | adantr 274 | . . . . . . . . 9 |
11 | simpl2 986 | . . . . . . . . . 10 | |
12 | 0z 9178 | . . . . . . . . . . . . 13 | |
13 | zapne 9238 | . . . . . . . . . . . . 13 # | |
14 | 12, 13 | mpan2 422 | . . . . . . . . . . . 12 # |
15 | 14 | 3ad2ant1 1003 | . . . . . . . . . . 11 # |
16 | 15 | adantr 274 | . . . . . . . . . 10 # |
17 | 11, 16 | mpbird 166 | . . . . . . . . 9 # |
18 | 5, 7, 10, 17 | divmulap3d 8698 | . . . . . . . 8 |
19 | eqcom 2159 | . . . . . . . 8 | |
20 | 18, 19 | bitrdi 195 | . . . . . . 7 |
21 | 20 | biimprd 157 | . . . . . 6 |
22 | 21 | impr 377 | . . . . 5 |
23 | simprl 521 | . . . . 5 | |
24 | 22, 23 | eqeltrd 2234 | . . . 4 |
25 | 24 | rexlimdvaa 2575 | . . 3 |
26 | simpr 109 | . . . . 5 | |
27 | simp2 983 | . . . . . . . 8 | |
28 | 27, 15 | mpbird 166 | . . . . . . 7 # |
29 | 4, 9, 28 | divcanap1d 8664 | . . . . . 6 |
30 | 29 | adantr 274 | . . . . 5 |
31 | oveq1 5831 | . . . . . . 7 | |
32 | 31 | eqeq1d 2166 | . . . . . 6 |
33 | 32 | rspcev 2816 | . . . . 5 |
34 | 26, 30, 33 | syl2anc 409 | . . . 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 963 wceq 1335 wcel 2128 wne 2327 wrex 2436 class class class wbr 3965 (class class class)co 5824 cc 7730 cc0 7732 cmul 7737 # cap 8456 cdiv 8545 cz 9167 cdvds 11683 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4253 df-po 4256 df-iso 4257 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-n0 9091 df-z 9168 df-dvds 11684 |
This theorem is referenced by: dvdsval3 11687 nndivdvds 11692 divconjdvds 11741 zeo3 11759 evend2 11780 oddp1d2 11781 fldivndvdslt 11826 divgcdz 11855 dvdsgcdidd 11878 mulgcd 11900 sqgcd 11913 lcmgcdlem 11954 mulgcddvds 11971 qredeu 11974 prmind2 11997 divgcdodd 12018 divnumden 12071 hashdvds 12096 hashgcdlem 12113 |
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