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Theorem prmuloc 7104
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 6947 . . 3 (𝐴 <Q 𝐵 → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵)
21adantl 271 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵)
3 prml 7015 . . . 4 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑟Q 𝑟𝐿)
43ad2antrr 472 . . 3 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) → ∃𝑟Q 𝑟𝐿)
5 simprl 498 . . . . . 6 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝑟Q)
6 simplrl 502 . . . . . 6 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝑥Q)
7 mulclnq 6914 . . . . . 6 ((𝑟Q𝑥Q) → (𝑟 ·Q 𝑥) ∈ Q)
85, 6, 7syl2anc 403 . . . . 5 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → (𝑟 ·Q 𝑥) ∈ Q)
9 ltrelnq 6903 . . . . . . . 8 <Q ⊆ (Q × Q)
109brel 4478 . . . . . . 7 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
1110simprd 112 . . . . . 6 (𝐴 <Q 𝐵𝐵Q)
1211ad3antlr 477 . . . . 5 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝐵Q)
13 appdiv0nq 7102 . . . . 5 (((𝑟 ·Q 𝑥) ∈ Q𝐵Q) → ∃𝑝Q (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))
148, 12, 13syl2anc 403 . . . 4 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → ∃𝑝Q (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))
15 prarloc 7041 . . . . . . . . . 10 ((⟨𝐿, 𝑈⟩ ∈ P𝑝Q) → ∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝))
1615adantlr 461 . . . . . . . . 9 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ 𝑝Q) → ∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝))
1716adantlr 461 . . . . . . . 8 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ 𝑝Q) → ∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝))
1817ad2ant2r 493 . . . . . . 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 2398 . . . . . . 7 (∃𝑑𝐿𝑢𝑈 𝑢 <Q (𝑑 +Q 𝑝) ↔ ∃𝑑𝑢((𝑑𝐿𝑢𝑈) ∧ 𝑢 <Q (𝑑 +Q 𝑝)))
2018, 19sylib 120 . . . . . 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 7019 . . . . . . . . . . . . . 14 ((⟨𝐿, 𝑈⟩ ∈ P𝑑𝐿) → 𝑑Q)
2221adantlr 461 . . . . . . . . . . . . 13 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ 𝑑𝐿) → 𝑑Q)
2322adantlr 461 . . . . . . . . . . . 12 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ 𝑑𝐿) → 𝑑Q)
2423adantlr 461 . . . . . . . . . . 11 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ 𝑑𝐿) → 𝑑Q)
2524ad2ant2r 493 . . . . . . . . . 10 ((((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑝Q ∧ (𝑝 ·Q 𝐵) <Q (𝑟 ·Q 𝑥))) ∧ (𝑑𝐿𝑢𝑈)) → 𝑑Q)
2625adantrr 463 . . . . . . . . 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 500 . . . . . . . . . . 11 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → ⟨𝐿, 𝑈⟩ ∈ P)
2827ad2antrr 472 . . . . . . . . . 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 498 . . . . . . . . . . 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 112 . . . . . . . . . 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 7020 . . . . . . . . . 10 ((⟨𝐿, 𝑈⟩ ∈ P𝑢𝑈) → 𝑢Q)
3228, 30, 31syl2anc 403 . . . . . . . . 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 7025 . . . . . . . . . . . . . . . . 17 ((⟨𝐿, 𝑈⟩ ∈ P𝑟𝐿𝑢𝑈) → 𝑟 <Q 𝑢)
34333adant1r 1167 . . . . . . . . . . . . . . . 16 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ 𝑟𝐿𝑢𝑈) → 𝑟 <Q 𝑢)
35343adant2l 1168 . . . . . . . . . . . . . . 15 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑟Q𝑟𝐿) ∧ 𝑢𝑈) → 𝑟 <Q 𝑢)
36353adant3l 1170 . . . . . . . . . . . . . 14 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑟Q𝑟𝐿) ∧ (𝑑𝐿𝑢𝑈)) → 𝑟 <Q 𝑢)
37363adant1r 1167 . . . . . . . . . . . . 13 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿) ∧ (𝑑𝐿𝑢𝑈)) → 𝑟 <Q 𝑢)
38373expa 1143 . . . . . . . . . . . 12 (((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) ∧ (𝑑𝐿𝑢𝑈)) → 𝑟 <Q 𝑢)
3938ad2ant2r 493 . . . . . . . . . . 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 499 . . . . . . . . . . 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 503 . . . . . . . . . . . 12 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → (𝐴 +Q 𝑥) = 𝐵)
4241ad2antrr 472 . . . . . . . . . . 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 503 . . . . . . . . . . 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 110 . . . . . . . . . . . . 13 (𝐴 <Q 𝐵𝐴Q)
4544ad3antlr 477 . . . . . . . . . . . 12 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → 𝐴Q)
4645ad2antrr 472 . . . . . . . . . . 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 472 . . . . . . . . . . 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 502 . . . . . . . . . . 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 472 . . . . . . . . . . 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 7103 . . . . . . . . . 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 926 . . . . . . . . . 10 ((𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)) ↔ ((𝑑𝐿𝑢𝑈) ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
5229, 50, 51sylanbrc 408 . . . . . . . . 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 302 . . . . . . . 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 113 . . . . . . 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 1810 . . . . . 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 2398 . . . . 5 (∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)) ↔ ∃𝑑𝑢((𝑑Q𝑢Q) ∧ (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵))))
5856, 57sylibr 132 . . . 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 2493 . . 3 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
604, 59rexlimddv 2493 . 2 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
612, 60rexlimddv 2493 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 102  w3a 924   = wceq 1289  wex 1426  wcel 1438  wrex 2360  cop 3444   class class class wbr 3837  (class class class)co 5634  Qcnq 6818   +Q cplq 6820   ·Q cmq 6821   <Q cltq 6823  Pcnp 6829
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004
This theorem is referenced by:  prmuloc2  7105  mullocpr  7109
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