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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 11422 | . . 3 | |
2 | 1 | 3adant2 985 | . 2 |
3 | zcn 9027 | . . . . . . . . . . 11 | |
4 | 3 | 3ad2ant3 989 | . . . . . . . . . 10 |
5 | 4 | adantr 274 | . . . . . . . . 9 |
6 | zcn 9027 | . . . . . . . . . 10 | |
7 | 6 | adantl 275 | . . . . . . . . 9 |
8 | zcn 9027 | . . . . . . . . . . 11 | |
9 | 8 | 3ad2ant1 987 | . . . . . . . . . 10 |
10 | 9 | adantr 274 | . . . . . . . . 9 |
11 | simpl2 970 | . . . . . . . . . 10 | |
12 | 0z 9033 | . . . . . . . . . . . . 13 | |
13 | zapne 9093 | . . . . . . . . . . . . 13 # | |
14 | 12, 13 | mpan2 421 | . . . . . . . . . . . 12 # |
15 | 14 | 3ad2ant1 987 | . . . . . . . . . . 11 # |
16 | 15 | adantr 274 | . . . . . . . . . 10 # |
17 | 11, 16 | mpbird 166 | . . . . . . . . 9 # |
18 | 5, 7, 10, 17 | divmulap3d 8553 | . . . . . . . 8 |
19 | eqcom 2119 | . . . . . . . 8 | |
20 | 18, 19 | syl6bb 195 | . . . . . . 7 |
21 | 20 | biimprd 157 | . . . . . 6 |
22 | 21 | impr 376 | . . . . 5 |
23 | simprl 505 | . . . . 5 | |
24 | 22, 23 | eqeltrd 2194 | . . . 4 |
25 | 24 | rexlimdvaa 2527 | . . 3 |
26 | simpr 109 | . . . . 5 | |
27 | simp2 967 | . . . . . . . 8 | |
28 | 27, 15 | mpbird 166 | . . . . . . 7 # |
29 | 4, 9, 28 | divcanap1d 8519 | . . . . . 6 |
30 | 29 | adantr 274 | . . . . 5 |
31 | oveq1 5749 | . . . . . . 7 | |
32 | 31 | eqeq1d 2126 | . . . . . 6 |
33 | 32 | rspcev 2763 | . . . . 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 947 wceq 1316 wcel 1465 wne 2285 wrex 2394 class class class wbr 3899 (class class class)co 5742 cc 7586 cc0 7588 cmul 7593 # cap 8311 cdiv 8400 cz 9022 cdvds 11420 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-n0 8946 df-z 9023 df-dvds 11421 |
This theorem is referenced by: dvdsval3 11424 nndivdvds 11426 divconjdvds 11474 zeo3 11492 evend2 11513 oddp1d2 11514 fldivndvdslt 11559 divgcdz 11587 dvdsgcdidd 11609 mulgcd 11631 sqgcd 11644 lcmgcdlem 11685 mulgcddvds 11702 qredeu 11705 prmind2 11728 divgcdodd 11748 divnumden 11801 hashdvds 11824 hashgcdlem 11830 |
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