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Mirrors > Home > ILE Home > Th. List > znq | Unicode version |
Description: The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.) |
Ref | Expression |
---|---|
znq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 |
. . 3
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2 | rspceov 5910 |
. . 3
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3 | 1, 2 | mp3an3 1326 |
. 2
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4 | elq 9598 |
. 2
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5 | 3, 4 | sylibr 134 |
1
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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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-po 4292 df-iso 4293 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-z 9230 df-q 9596 |
This theorem is referenced by: qnegcl 9612 qreccl 9618 nnrecq 9621 elpqb 9625 qbtwnre 10230 adddivflid 10265 fldivnn0 10268 divfl0 10269 flhalf 10275 fldivnn0le 10276 flltdivnn0lt 10277 fldiv4p1lem1div2 10278 intfracq 10293 flqdiv 10294 zmodcl 10317 iexpcyc 10597 facavg 10697 bcval 10700 eirraplem 11755 dvdsmod 11838 divalglemnn 11893 divalgmod 11902 flodddiv4 11909 flodddiv4t2lthalf 11912 modgcd 11962 qredeu 12067 sqrt2irraplemnn 12149 sqrt2irrap 12150 divnumden 12166 hashdvds 12191 prmdiv 12205 phisum 12210 odzdvds 12215 pcdiv 12272 pcaddlem 12308 pcmptdvds 12313 fldivp1 12316 pcfaclem 12317 pcfac 12318 pcbc 12319 4sqlem5 12350 4sqlem6 12351 4sqlem10 12355 mulgmodid 12897 logbgcd1irraplemap 14020 ex-fl 14099 ex-ceil 14100 |
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