![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > znq | GIF version |
Description: The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.) |
Ref | Expression |
---|---|
znq | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2187 | . . 3 ⊢ (𝐴 / 𝐵) = (𝐴 / 𝐵) | |
2 | rspceov 5930 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ ∧ (𝐴 / 𝐵) = (𝐴 / 𝐵)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝐴 / 𝐵) = (𝑥 / 𝑦)) | |
3 | 1, 2 | mp3an3 1336 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝐴 / 𝐵) = (𝑥 / 𝑦)) |
4 | elq 9635 | . 2 ⊢ ((𝐴 / 𝐵) ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝐴 / 𝐵) = (𝑥 / 𝑦)) | |
5 | 3, 4 | sylibr 134 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 ∃wrex 2466 (class class class)co 5888 / cdiv 8642 ℕcn 8932 ℤcz 9266 ℚcq 9632 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-po 4308 df-iso 4309 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-z 9267 df-q 9633 |
This theorem is referenced by: qnegcl 9649 qreccl 9655 nnrecq 9658 elpqb 9662 qbtwnre 10270 adddivflid 10305 fldivnn0 10308 divfl0 10309 flhalf 10315 fldivnn0le 10316 flltdivnn0lt 10317 fldiv4p1lem1div2 10318 intfracq 10333 flqdiv 10334 zmodcl 10357 iexpcyc 10638 facavg 10739 bcval 10742 eirraplem 11797 dvdsmod 11881 divalglemnn 11936 divalgmod 11945 flodddiv4 11952 flodddiv4t2lthalf 11955 modgcd 12005 qredeu 12110 sqrt2irraplemnn 12192 sqrt2irrap 12193 divnumden 12209 hashdvds 12234 prmdiv 12248 phisum 12253 odzdvds 12258 pcdiv 12315 pcaddlem 12351 pcmptdvds 12356 fldivp1 12359 pcfaclem 12360 pcfac 12361 pcbc 12362 4sqlem5 12393 4sqlem6 12394 4sqlem10 12398 mulgmodid 13053 logbgcd1irraplemap 14658 lgseisenlem2 14722 ex-fl 14748 ex-ceil 14749 |
Copyright terms: Public domain | W3C validator |