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Theorem prmuloc 7556
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 7399 . . 3 (𝐴 <Q 𝐵 → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵)
21adantl 277 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵)
3 prml 7467 . . . 4 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑟Q 𝑟𝐿)
43ad2antrr 488 . . 3 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) → ∃𝑟Q 𝑟𝐿)
5 simprl 529 . . . . . 6 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝑟Q)
6 simplrl 535 . . . . . 6 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝑥Q)
7 mulclnq 7366 . . . . . 6 ((𝑟Q𝑥Q) → (𝑟 ·Q 𝑥) ∈ Q)
85, 6, 7syl2anc 411 . . . . 5 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → (𝑟 ·Q 𝑥) ∈ Q)
9 ltrelnq 7355 . . . . . . . 8 <Q ⊆ (Q × Q)
109brel 4675 . . . . . . 7 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
1110simprd 114 . . . . . 6 (𝐴 <Q 𝐵𝐵Q)
1211ad3antlr 493 . . . . 5 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝐵Q)
13 appdiv0nq 7554 . . . . 5 (((𝑟 ·Q 𝑥) ∈ Q𝐵Q) → ∃𝑝Q (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))
148, 12, 13syl2anc 411 . . . 4 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → ∃𝑝Q (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))
15 prarloc 7493 . . . . . . . . . 10 ((⟨𝐿, 𝑈⟩ ∈ P𝑝Q) → ∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝))
1615adantlr 477 . . . . . . . . 9 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ 𝑝Q) → ∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝))
1716adantlr 477 . . . . . . . 8 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ 𝑝Q) → ∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝))
1817ad2ant2r 509 . . . . . . 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 2497 . . . . . . 7 (∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝) ↔ ∃𝑑𝑢((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝)))
2018, 19sylib 122 . . . . . 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 7471 . . . . . . . . . . . . . 14 ((⟨𝐿, 𝑈⟩ ∈ P𝑑𝐿) → 𝑑Q)
2221adantlr 477 . . . . . . . . . . . . 13 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ 𝑑𝐿) → 𝑑Q)
2322adantlr 477 . . . . . . . . . . . 12 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ 𝑑𝐿) → 𝑑Q)
2423adantlr 477 . . . . . . . . . . 11 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ 𝑑𝐿) → 𝑑Q)
2524ad2ant2r 509 . . . . . . . . . 10 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ (𝑑𝐿𝑢𝑈)) → 𝑑Q)
2625adantrr 479 . . . . . . . . 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 533 . . . . . . . . . . 11 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → ⟨𝐿, 𝑈⟩ ∈ P)
2827ad2antrr 488 . . . . . . . . . 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 529 . . . . . . . . . . 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 114 . . . . . . . . . 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 7472 . . . . . . . . . 10 ((⟨𝐿, 𝑈⟩ ∈ P𝑢𝑈) → 𝑢Q)
3228, 30, 31syl2anc 411 . . . . . . . . 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 7477 . . . . . . . . . . . . . . . . 17 ((⟨𝐿, 𝑈⟩ ∈ P𝑟𝐿𝑢𝑈) → 𝑟 <Q 𝑢)
34333adant1r 1231 . . . . . . . . . . . . . . . 16 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ 𝑟𝐿𝑢𝑈) → 𝑟 <Q 𝑢)
35343adant2l 1232 . . . . . . . . . . . . . . 15 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑟Q𝑟𝐿) ∧ 𝑢𝑈) → 𝑟 <Q 𝑢)
36353adant3l 1234 . . . . . . . . . . . . . 14 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑟Q𝑟𝐿) ∧ (𝑑𝐿𝑢𝑈)) → 𝑟 <Q 𝑢)
37363adant1r 1231 . . . . . . . . . . . . 13 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿) ∧ (𝑑𝐿𝑢𝑈)) → 𝑟 <Q 𝑢)
38373expa 1203 . . . . . . . . . . . 12 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑑𝐿𝑢𝑈)) → 𝑟 <Q 𝑢)
3938ad2ant2r 509 . . . . . . . . . . 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 531 . . . . . . . . . . 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 536 . . . . . . . . . . . 12 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → (𝐴 +Q 𝑥) = 𝐵)
4241ad2antrr 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))) → (𝐴 +Q 𝑥) = 𝐵)
43 simplrr 536 . . . . . . . . . . 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 112 . . . . . . . . . . . . 13 (𝐴 <Q 𝐵𝐴Q)
4544ad3antlr 493 . . . . . . . . . . . 12 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝐴Q)
4645ad2antrr 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))) → 𝐴Q)
4712ad2antrr 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))) → 𝐵Q)
48 simplrl 535 . . . . . . . . . . 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 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))) → 𝑥Q)
5039, 40, 42, 43, 46, 47, 26, 48, 49prmuloclemcalc 7555 . . . . . . . . . 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 980 . . . . . . . . . 10 ((𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)) ↔ ((𝑑𝐿𝑢𝑈) ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
5229, 50, 51sylanbrc 417 . . . . . . . . 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 309 . . . . . . . 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 115 . . . . . . 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 1882 . . . . . 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 2497 . . . . 5 (∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)) ↔ ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵))))
5856, 57sylibr 134 . . . 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 2599 . . 3 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
604, 59rexlimddv 2599 . 2 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
612, 60rexlimddv 2599 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 104  w3a 978   = wceq 1353  wex 1492  wcel 2148  wrex 2456  cop 3594   class class class wbr 4000  (class class class)co 5869  Qcnq 7270   +Q cplq 7272   ·Q cmq 7273   <Q cltq 7275  Pcnp 7281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456
This theorem is referenced by:  prmuloc2  7557  mullocpr  7561
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