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

Theorem prmuloc 7028
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 6871 . . 3 (𝐴 <Q 𝐵 → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵)
21adantl 271 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) → ∃𝑥Q (𝐴 +Q 𝑥) = 𝐵)
3 prml 6939 . . . 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 6838 . . . . . 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 6827 . . . . . . . 8 <Q ⊆ (Q × Q)
109brel 4448 . . . . . . 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 7026 . . . . 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 6965 . . . . . . . . . 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 2392 . . . . . . 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 6943 . . . . . . . . . . . . . 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 6944 . . . . . . . . . 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 6949 . . . . . . . . . . . . . . . . 17 ((⟨𝐿, 𝑈⟩ ∈ P𝑟𝐿𝑢𝑈) → 𝑟 <Q 𝑢)
34333adant1r 1163 . . . . . . . . . . . . . . . 16 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ 𝑟𝐿𝑢𝑈) → 𝑟 <Q 𝑢)
35343adant2l 1164 . . . . . . . . . . . . . . 15 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑟Q𝑟𝐿) ∧ 𝑢𝑈) → 𝑟 <Q 𝑢)
36353adant3l 1166 . . . . . . . . . . . . . 14 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑟Q𝑟𝐿) ∧ (𝑑𝐿𝑢𝑈)) → 𝑟 <Q 𝑢)
37363adant1r 1163 . . . . . . . . . . . . 13 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿) ∧ (𝑑𝐿𝑢𝑈)) → 𝑟 <Q 𝑢)
38373expa 1139 . . . . . . . . . . . 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 7027 . . . . . . . . . 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 922 . . . . . . . . . 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 1805 . . . . . 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 2392 . . . . 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 2487 . . 3 ((((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) ∧ (𝑟Q𝑟𝐿)) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
604, 59rexlimddv 2487 . 2 (((⟨𝐿, 𝑈⟩ ∈ P𝐴 <Q 𝐵) ∧ (𝑥Q ∧ (𝐴 +Q 𝑥) = 𝐵)) → ∃𝑑Q𝑢Q (𝑑𝐿𝑢𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
612, 60rexlimddv 2487 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 920   = wceq 1285  wex 1422  wcel 1434  wrex 2354  cop 3425   class class class wbr 3811  (class class class)co 5591  Qcnq 6742   +Q cplq 6744   ·Q cmq 6745   <Q cltq 6747  Pcnp 6753
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928
This theorem is referenced by:  prmuloc2  7029  mullocpr  7033
  Copyright terms: Public domain W3C validator