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Theorem appdiv0nq 7562
Description: Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7561 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 7410 . . 3 (𝐵Q → ∃𝑥Q 𝑥 <Q 𝐵)
21adantr 276 . 2 ((𝐵Q𝐶Q) → ∃𝑥Q 𝑥 <Q 𝐵)
3 appdivnq 7561 . . . . 5 ((𝑥 <Q 𝐵𝐶Q) → ∃𝑚Q (𝑥 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵))
4 simpr 110 . . . . . 6 ((𝑥 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵) → (𝑚 ·Q 𝐶) <Q 𝐵)
54reximi 2574 . . . . 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 268 . . 3 ((𝐶Q𝑥 <Q 𝐵) → ∃𝑚Q (𝑚 ·Q 𝐶) <Q 𝐵)
87ad2ant2l 508 . 2 (((𝐵Q𝐶Q) ∧ (𝑥Q𝑥 <Q 𝐵)) → ∃𝑚Q (𝑚 ·Q 𝐶) <Q 𝐵)
92, 8rexlimddv 2599 1 ((𝐵Q𝐶Q) → ∃𝑚Q (𝑚 ·Q 𝐶) <Q 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2148  wrex 2456   class class class wbr 4003  (class class class)co 5874  Qcnq 7278   ·Q cmq 7281   <Q cltq 7283
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-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  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-ral 2460  df-rex 2461  df-reu 2462  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-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351
This theorem is referenced by:  prmuloc  7564
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