ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  appdiv0nq GIF version

Theorem appdiv0nq 7392
Description: Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7391 in which 𝐴 is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.)
Assertion
Ref Expression
appdiv0nq ((𝐵Q𝐶Q) → ∃𝑚Q (𝑚 ·Q 𝐶) <Q 𝐵)
Distinct variable groups:   𝐵,𝑚   𝐶,𝑚

Proof of Theorem appdiv0nq
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nsmallnqq 7240 . . 3 (𝐵Q → ∃𝑥Q 𝑥 <Q 𝐵)
21adantr 274 . 2 ((𝐵Q𝐶Q) → ∃𝑥Q 𝑥 <Q 𝐵)
3 appdivnq 7391 . . . . 5 ((𝑥 <Q 𝐵𝐶Q) → ∃𝑚Q (𝑥 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵))
4 simpr 109 . . . . . 6 ((𝑥 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵) → (𝑚 ·Q 𝐶) <Q 𝐵)
54reximi 2530 . . . . 5 (∃𝑚Q (𝑥 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵) → ∃𝑚Q (𝑚 ·Q 𝐶) <Q 𝐵)
63, 5syl 14 . . . 4 ((𝑥 <Q 𝐵𝐶Q) → ∃𝑚Q (𝑚 ·Q 𝐶) <Q 𝐵)
76ancoms 266 . . 3 ((𝐶Q𝑥 <Q 𝐵) → ∃𝑚Q (𝑚 ·Q 𝐶) <Q 𝐵)
87ad2ant2l 500 . 2 (((𝐵Q𝐶Q) ∧ (𝑥Q𝑥 <Q 𝐵)) → ∃𝑚Q (𝑚 ·Q 𝐶) <Q 𝐵)
92, 8rexlimddv 2555 1 ((𝐵Q𝐶Q) → ∃𝑚Q (𝑚 ·Q 𝐶) <Q 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1481  wrex 2418   class class class wbr 3933  (class class class)co 5778  Qcnq 7108   ·Q cmq 7111   <Q cltq 7113
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-eprel 4215  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-irdg 6271  df-1o 6317  df-oadd 6321  df-omul 6322  df-er 6433  df-ec 6435  df-qs 6439  df-ni 7132  df-pli 7133  df-mi 7134  df-lti 7135  df-plpq 7172  df-mpq 7173  df-enq 7175  df-nqqs 7176  df-plqqs 7177  df-mqqs 7178  df-1nqqs 7179  df-rq 7180  df-ltnqqs 7181
This theorem is referenced by:  prmuloc  7394
  Copyright terms: Public domain W3C validator