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Theorem prarloc2 7254
Description: A Dedekind cut is arithmetically located. This is a variation of prarloc 7253 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance 𝑃, there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.)
Assertion
Ref Expression
prarloc2 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑎𝐿 (𝑎 +Q 𝑃) ∈ 𝑈)
Distinct variable groups:   𝐿,𝑎   𝑃,𝑎   𝑈,𝑎

Proof of Theorem prarloc2
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 prarloc 7253 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑎𝐿𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃))
2 prcunqu 7235 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝑏𝑈) → (𝑏 <Q (𝑎 +Q 𝑃) → (𝑎 +Q 𝑃) ∈ 𝑈))
32rexlimdva 2521 . . . 4 (⟨𝐿, 𝑈⟩ ∈ P → (∃𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃) → (𝑎 +Q 𝑃) ∈ 𝑈))
43reximdv 2505 . . 3 (⟨𝐿, 𝑈⟩ ∈ P → (∃𝑎𝐿𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃) → ∃𝑎𝐿 (𝑎 +Q 𝑃) ∈ 𝑈))
54adantr 272 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → (∃𝑎𝐿𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃) → ∃𝑎𝐿 (𝑎 +Q 𝑃) ∈ 𝑈))
61, 5mpd 13 1 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑎𝐿 (𝑎 +Q 𝑃) ∈ 𝑈)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1461  wrex 2389  cop 3494   class class class wbr 3893  (class class class)co 5726  Qcnq 7030   +Q cplq 7032   <Q cltq 7035  Pcnp 7041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-2o 6266  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103  df-enq0 7174  df-nq0 7175  df-0nq0 7176  df-plq0 7177  df-mq0 7178  df-inp 7216
This theorem is referenced by:  addcanprleml  7364  addcanprlemu  7365  aptiprleml  7389  aptiprlemu  7390
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