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Theorem prmuloc 6664
 Description: Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.)
Assertion
Ref Expression
prmuloc ((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
Distinct variable groups:   𝐴,𝑑,𝑢   𝐵,𝑑,𝑢   𝐿,𝑑,𝑢   𝑈,𝑑,𝑢

Proof of Theorem prmuloc
Dummy variables 𝑝 𝑟 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqi 6507 . . 3 (𝐴 <Q 𝐵 → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵)
21adantl 262 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵)
3 prml 6575 . . . 4 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑟Q 𝑟𝐿)
43ad2antrr 457 . . 3 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) → ∃𝑟Q 𝑟𝐿)
5 simprl 483 . . . . . 6 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝑟Q)
6 simplrl 487 . . . . . 6 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝑥Q)
7 mulclnq 6474 . . . . . 6 ((𝑟Q𝑥Q) → (𝑟 ·Q 𝑥) ∈ Q)
85, 6, 7syl2anc 391 . . . . 5 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → (𝑟 ·Q 𝑥) ∈ Q)
9 ltrelnq 6463 . . . . . . . 8 <Q ⊆ (Q × Q)
109brel 4392 . . . . . . 7 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
1110simprd 107 . . . . . 6 (𝐴 <Q 𝐵𝐵Q)
1211ad3antlr 462 . . . . 5 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝐵Q)
13 appdiv0nq 6662 . . . . 5 (((𝑟 ·Q 𝑥) ∈ Q𝐵Q) → ∃𝑝Q (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))
148, 12, 13syl2anc 391 . . . 4 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → ∃𝑝Q (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))
15 prarloc 6601 . . . . . . . . . 10 ((⟨𝐿, 𝑈⟩ ∈ P𝑝Q) → ∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝))
1615adantlr 446 . . . . . . . . 9 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ 𝑝Q) → ∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝))
1716adantlr 446 . . . . . . . 8 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ 𝑝Q) → ∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝))
1817ad2ant2r 478 . . . . . . 7 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) → ∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝))
19 r2ex 2344 . . . . . . 7 (∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝) ↔ ∃𝑑𝑢((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝)))
2018, 19sylib 127 . . . . . 6 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) → ∃𝑑𝑢((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝)))
21 elprnql 6579 . . . . . . . . . . . . . 14 ((⟨𝐿, 𝑈⟩ ∈ P𝑑𝐿) → 𝑑Q)
2221adantlr 446 . . . . . . . . . . . . 13 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ 𝑑𝐿) → 𝑑Q)
2322adantlr 446 . . . . . . . . . . . 12 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ 𝑑𝐿) → 𝑑Q)
2423adantlr 446 . . . . . . . . . . 11 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ 𝑑𝐿) → 𝑑Q)
2524ad2ant2r 478 . . . . . . . . . 10 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ (𝑑𝐿𝑢𝑈)) → 𝑑Q)
2625adantrr 448 . . . . . . . . 9 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → 𝑑Q)
27 simplll 485 . . . . . . . . . . 11 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → ⟨𝐿, 𝑈⟩ ∈ P)
2827ad2antrr 457 . . . . . . . . . 10 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → ⟨𝐿, 𝑈⟩ ∈ P)
29 simprl 483 . . . . . . . . . . 11 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → (𝑑𝐿𝑢𝑈))
3029simprd 107 . . . . . . . . . 10 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → 𝑢𝑈)
31 elprnqu 6580 . . . . . . . . . 10 ((⟨𝐿, 𝑈⟩ ∈ P𝑢𝑈) → 𝑢Q)
3228, 30, 31syl2anc 391 . . . . . . . . 9 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → 𝑢Q)
33 prltlu 6585 . . . . . . . . . . . . . . . . 17 ((⟨𝐿, 𝑈⟩ ∈ P𝑟𝐿𝑢𝑈) → 𝑟 <Q 𝑢)
34333adant1r 1128 . . . . . . . . . . . . . . . 16 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ 𝑟𝐿𝑢𝑈) → 𝑟 <Q 𝑢)
35343adant2l 1129 . . . . . . . . . . . . . . 15 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑟Q𝑟𝐿) ∧ 𝑢𝑈) → 𝑟 <Q 𝑢)
36353adant3l 1131 . . . . . . . . . . . . . 14 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑟Q𝑟𝐿) ∧ (𝑑𝐿𝑢𝑈)) → 𝑟 <Q 𝑢)
37363adant1r 1128 . . . . . . . . . . . . 13 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿) ∧ (𝑑𝐿𝑢𝑈)) → 𝑟 <Q 𝑢)
38373expa 1104 . . . . . . . . . . . 12 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑑𝐿𝑢𝑈)) → 𝑟 <Q 𝑢)
3938ad2ant2r 478 . . . . . . . . . . 11 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → 𝑟 <Q 𝑢)
40 simprr 484 . . . . . . . . . . 11 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → 𝑢 <Q (𝑑 +Q 𝑝))
41 simplrr 488 . . . . . . . . . . . 12 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → (𝐴 +Q 𝑥) = 𝐵)
4241ad2antrr 457 . . . . . . . . . . 11 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → (𝐴 +Q 𝑥) = 𝐵)
43 simplrr 488 . . . . . . . . . . 11 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))
4410simpld 105 . . . . . . . . . . . . 13 (𝐴 <Q 𝐵𝐴Q)
4544ad3antlr 462 . . . . . . . . . . . 12 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝐴Q)
4645ad2antrr 457 . . . . . . . . . . 11 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → 𝐴Q)
4712ad2antrr 457 . . . . . . . . . . 11 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → 𝐵Q)
48 simplrl 487 . . . . . . . . . . 11 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → 𝑝Q)
496ad2antrr 457 . . . . . . . . . . 11 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → 𝑥Q)
5039, 40, 42, 43, 46, 47, 26, 48, 49prmuloclemcalc 6663 . . . . . . . . . 10 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵))
51 df-3an 887 . . . . . . . . . 10 ((𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)) ↔ ((𝑑𝐿𝑢𝑈) ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
5229, 50, 51sylanbrc 394 . . . . . . . . 9 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
5326, 32, 52jca31 292 . . . . . . . 8 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ ((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝))) → ((𝑑Q𝑢Q) ∧ (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵))))
5453ex 108 . . . . . . 7 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) → (((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝)) → ((𝑑Q𝑢Q) ∧ (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))))
55542eximdv 1762 . . . . . 6 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) → (∃𝑑𝑢((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝)) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))))
5620, 55mpd 13 . . . . 5 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) → ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵))))
57 r2ex 2344 . . . . 5 (∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)) ↔ ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵))))
5856, 57sylibr 137 . . . 4 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
5914, 58rexlimddv 2437 . . 3 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
604, 59rexlimddv 2437 . 2 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
612, 60rexlimddv 2437 1 ((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378   class class class wbr 3764  (class class class)co 5512  Qcnq 6378   +Q cplq 6380   ·Q cmq 6381
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