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Theorem prmuloc2 7419
 Description: Positive reals are multiplicatively located. This is a variation of prmuloc 7418 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 7418 . 2 ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥Q𝑦Q (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)))
2 nfv 1509 . . 3 𝑥(⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵)
3 nfre1 2480 . . 3 𝑥𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈
4 simpr1 988 . . . . . . . 8 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑥𝐿)
5 simpr3 990 . . . . . . . . . 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 526 . . . . . . . . . . 11 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑦Q)
7 mulidnq 7241 . . . . . . . . . . 11 (𝑦Q → (𝑦 ·Q 1Q) = 𝑦)
8 breq1 3941 . . . . . . . . . . 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 146 . . . . . . . . 9 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑦 <Q (𝑥 ·Q 𝐵))
11 simplll 523 . . . . . . . . . 10 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → ⟨𝐿, 𝑈⟩ ∈ P)
12 simpr2 989 . . . . . . . . . 10 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → 𝑦𝑈)
13 prcunqu 7337 . . . . . . . . . 10 ((⟨𝐿, 𝑈⟩ ∈ P𝑦𝑈) → (𝑦 <Q (𝑥 ·Q 𝐵) → (𝑥 ·Q 𝐵) ∈ 𝑈))
1411, 12, 13syl2anc 409 . . . . . . . . 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 2485 . . . . . . . 8 ((𝑥𝐿 ∧ (𝑥 ·Q 𝐵) ∈ 𝑈) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
174, 15, 16syl2anc 409 . . . . . . 7 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) ∧ (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵))) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
1817ex 114 . . . . . 6 (((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ (𝑥Q𝑦Q)) → ((𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈))
1918anassrs 398 . . . . 5 ((((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ 𝑥Q) ∧ 𝑦Q) → ((𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈))
2019rexlimdva 2553 . . . 4 (((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) ∧ 𝑥Q) → (∃𝑦Q (𝑥𝐿𝑦𝑈 ∧ (𝑦 ·Q 1Q) <Q (𝑥 ·Q 𝐵)) → ∃𝑥𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈))
2120ex 114 . . 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 2550 . 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 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∃wrex 2418  ⟨cop 3536   class class class wbr 3938  (class class class)co 5783  Qcnq 7132  1Qc1q 7133   ·Q cmq 7135
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