| Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 12019 |
. . 3
| |
| 2 | 1 | 3adant2 1018 |
. 2
|
| 3 | zcn 9359 |
. . . . . . . . . . 11
| |
| 4 | 3 | 3ad2ant3 1022 |
. . . . . . . . . 10
|
| 5 | 4 | adantr 276 |
. . . . . . . . 9
|
| 6 | zcn 9359 |
. . . . . . . . . 10
| |
| 7 | 6 | adantl 277 |
. . . . . . . . 9
|
| 8 | zcn 9359 |
. . . . . . . . . . 11
| |
| 9 | 8 | 3ad2ant1 1020 |
. . . . . . . . . 10
|
| 10 | 9 | adantr 276 |
. . . . . . . . 9
|
| 11 | simpl2 1003 |
. . . . . . . . . 10
| |
| 12 | 0z 9365 |
. . . . . . . . . . . . 13
| |
| 13 | zapne 9429 |
. . . . . . . . . . . . 13
| |
| 14 | 12, 13 | mpan2 425 |
. . . . . . . . . . . 12
|
| 15 | 14 | 3ad2ant1 1020 |
. . . . . . . . . . 11
|
| 16 | 15 | adantr 276 |
. . . . . . . . . 10
|
| 17 | 11, 16 | mpbird 167 |
. . . . . . . . 9
|
| 18 | 5, 7, 10, 17 | divmulap3d 8880 |
. . . . . . . 8
|
| 19 | eqcom 2206 |
. . . . . . . 8
| |
| 20 | 18, 19 | bitrdi 196 |
. . . . . . 7
|
| 21 | 20 | biimprd 158 |
. . . . . 6
|
| 22 | 21 | impr 379 |
. . . . 5
|
| 23 | simprl 529 |
. . . . 5
| |
| 24 | 22, 23 | eqeltrd 2281 |
. . . 4
|
| 25 | 24 | rexlimdvaa 2623 |
. . 3
|
| 26 | simpr 110 |
. . . . 5
| |
| 27 | simp2 1000 |
. . . . . . . 8
| |
| 28 | 27, 15 | mpbird 167 |
. . . . . . 7
|
| 29 | 4, 9, 28 | divcanap1d 8846 |
. . . . . 6
|
| 30 | 29 | adantr 276 |
. . . . 5
|
| 31 | oveq1 5941 |
. . . . . . 7
| |
| 32 | 31 | eqeq1d 2213 |
. . . . . 6
|
| 33 | 32 | rspcev 2876 |
. . . . 5
|
| 34 | 26, 30, 33 | syl2anc 411 |
. . . 4
|
| 35 | 34 | ex 115 |
. . 3
|
| 36 | 25, 35 | impbid 129 |
. 2
|
| 37 | 2, 36 | bitrd 188 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-mulrcl 8006 ax-addcom 8007 ax-mulcom 8008 ax-addass 8009 ax-mulass 8010 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-1rid 8014 ax-0id 8015 ax-rnegex 8016 ax-precex 8017 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-apti 8022 ax-pre-ltadd 8023 ax-pre-mulgt0 8024 ax-pre-mulext 8025 |
| This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-br 4044 df-opab 4105 df-id 4338 df-po 4341 df-iso 4342 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-iota 5229 df-fun 5270 df-fv 5276 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-reap 8630 df-ap 8637 df-div 8728 df-inn 9019 df-n0 9278 df-z 9355 df-dvds 12018 |
| This theorem is referenced by: dvdsval3 12021 nndivdvds 12026 fsumdvds 12072 divconjdvds 12079 3dvds 12094 zeo3 12098 evend2 12119 oddp1d2 12120 fldivndvdslt 12167 bitsmod 12186 divgcdz 12211 dvdsgcdidd 12234 mulgcd 12256 sqgcd 12269 lcmgcdlem 12318 mulgcddvds 12335 qredeu 12338 prmind2 12361 isprm5lem 12382 divgcdodd 12384 divnumden 12437 hashdvds 12462 hashgcdlem 12479 pythagtriplem19 12524 pcprendvds2 12533 pcpremul 12535 pc2dvds 12572 pcz 12574 dvdsprmpweqle 12579 pcadd 12582 pcmptdvds 12587 fldivp1 12590 pockthlem 12598 4sqlem8 12627 4sqlem9 12628 4sqlem12 12644 4sqlem14 12646 znidomb 14338 lgseisenlem1 15465 lgsquad2lem1 15476 lgsquad3 15479 m1lgs 15480 2sqlem3 15512 2sqlem8 15518 |
| Copyright terms: Public domain | W3C validator |