ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prmuloc2 GIF version

Theorem prmuloc2 7047
Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7046 which only constructs one (named) point and is therefore often easier to work with. It states that given a ratio 𝐵, there are elements of the lower and upper cut which have exactly that ratio between them. (Contributed by Jim Kingdon, 28-Dec-2019.)
Assertion
Ref Expression
prmuloc2 ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐿   𝑥,𝑈

Proof of Theorem prmuloc2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 prmuloc 7046 . 2 ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥Q𝑦Q (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)))
2 nfv 1464 . . 3 𝑥(⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵)
3 nfre1 2415 . . 3 𝑥𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈
4 simpr1 947 . . . . . . . 8 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑥𝐿)
5 simpr3 949 . . . . . . . . . 10 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))
6 simplrr 503 . . . . . . . . . . 11 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑦Q)
7 mulidnq 6869 . . . . . . . . . . 11 (𝑦Q → (𝑦 ·Q 1Q) = 𝑦)
8 breq1 3817 . . . . . . . . . . 11 ((𝑦 ·Q 1Q) = 𝑦 → ((𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵) ↔ 𝑦 <Q (𝑥 ·Q 𝐵)))
96, 7, 83syl 17 . . . . . . . . . 10 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → ((𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵) ↔ 𝑦 <Q (𝑥 ·Q 𝐵)))
105, 9mpbid 145 . . . . . . . . 9 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑦 <Q (𝑥 ·Q 𝐵))
11 simplll 500 . . . . . . . . . 10 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → ⟨𝐿, 𝑈⟩ ∈ P)
12 simpr2 948 . . . . . . . . . 10 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑦𝑈)
13 prcunqu 6965 . . . . . . . . . 10 ((⟨𝐿, 𝑈⟩ ∈ P𝑦𝑈) → (𝑦 <Q (𝑥 ·Q 𝐵) → (𝑥 ·Q 𝐵) ∈ 𝑈))
1411, 12, 13syl2anc 403 . . . . . . . . 9 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → (𝑦 <Q (𝑥 ·Q 𝐵) → (𝑥 ·Q 𝐵) ∈ 𝑈))
1510, 14mpd 13 . . . . . . . 8 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → (𝑥 ·Q 𝐵) ∈ 𝑈)
16 rspe 2420 . . . . . . . 8 ((𝑥𝐿 ∧ (𝑥 ·Q 𝐵) ∈ 𝑈) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
174, 15, 16syl2anc 403 . . . . . . 7 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
1817ex 113 . . . . . 6 (((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) → ((𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈))
1918anassrs 392 . . . . 5 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ 𝑥Q) ∧ 𝑦Q) → ((𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈))
2019rexlimdva 2485 . . . 4 (((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ 𝑥Q) → (∃𝑦Q (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈))
2120ex 113 . . 3 ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → (𝑥Q → (∃𝑦Q (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)))
222, 3, 21rexlimd 2482 . 2 ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → (∃𝑥Q𝑦Q (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈))
231, 22mpd 13 1 ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 922   = wceq 1287  wcel 1436  wrex 2356  cop 3428   class class class wbr 3814  (class class class)co 5594  Qcnq 6760  1Qc1q 6761   ·Q cmq 6763   <Q cltq 6765  Pcnp 6771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-eprel 4083  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-irdg 6070  df-1o 6116  df-2o 6117  df-oadd 6120  df-omul 6121  df-er 6225  df-ec 6227  df-qs 6231  df-ni 6784  df-pli 6785  df-mi 6786  df-lti 6787  df-plpq 6824  df-mpq 6825  df-enq 6827  df-nqqs 6828  df-plqqs 6829  df-mqqs 6830  df-1nqqs 6831  df-rq 6832  df-ltnqqs 6833  df-enq0 6904  df-nq0 6905  df-0nq0 6906  df-plq0 6907  df-mq0 6908  df-inp 6946
This theorem is referenced by:  recexprlem1ssl  7113  recexprlem1ssu  7114
  Copyright terms: Public domain W3C validator