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Theorem prarloc2 6602
Description: A Dedekind cut is arithmetically located. This is a variation of prarloc 6601 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 6601 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑎𝐿𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃))
2 prcunqu 6583 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝑏𝑈) → (𝑏 <Q (𝑎 +Q 𝑃) → (𝑎 +Q 𝑃) ∈ 𝑈))
32rexlimdva 2433 . . . 4 (⟨𝐿, 𝑈⟩ ∈ P → (∃𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃) → (𝑎 +Q 𝑃) ∈ 𝑈))
43reximdv 2420 . . 3 (⟨𝐿, 𝑈⟩ ∈ P → (∃𝑎𝐿𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃) → ∃𝑎𝐿 (𝑎 +Q 𝑃) ∈ 𝑈))
54adantr 261 . 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 97  wcel 1393  wrex 2307  cop 3378   class class class wbr 3764  (class class class)co 5512  Qcnq 6378   +Q cplq 6380   <Q cltq 6383  Pcnp 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564
This theorem is referenced by:  addcanprleml  6712  addcanprlemu  6713  aptiprleml  6737  aptiprlemu  6738
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