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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 2189 |
. . 3
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2 | rspceov 5938 |
. . 3
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3 | 1, 2 | mp3an3 1337 |
. 2
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4 | elq 9652 |
. 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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 df-inn 8950 df-z 9284 df-q 9650 |
This theorem is referenced by: qnegcl 9666 qreccl 9672 nnrecq 9675 elpqb 9679 qbtwnre 10287 adddivflid 10323 fldivnn0 10326 divfl0 10327 flhalf 10333 fldivnn0le 10334 flltdivnn0lt 10335 fldiv4p1lem1div2 10336 intfracq 10351 flqdiv 10352 zmodcl 10375 iexpcyc 10656 facavg 10758 bcval 10761 eirraplem 11816 dvdsmod 11900 divalglemnn 11955 divalgmod 11964 flodddiv4 11971 flodddiv4t2lthalf 11974 modgcd 12024 qredeu 12129 sqrt2irraplemnn 12211 sqrt2irrap 12212 divnumden 12228 hashdvds 12253 prmdiv 12267 phisum 12272 odzdvds 12277 pcdiv 12334 pcaddlem 12371 pcmptdvds 12377 fldivp1 12380 pcfaclem 12381 pcfac 12382 pcbc 12383 4sqlem5 12414 4sqlem6 12415 4sqlem10 12419 mulgmodid 13101 logbgcd1irraplemap 14844 lgseisenlem2 14909 ex-fl 14935 ex-ceil 14936 |
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