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Theorem prarloc 7465
Description: A Dedekind cut is arithmetically located. Part of Proposition 11.15 of [BauerTaylor], p. 52, slightly modified. It states that given a tolerance 𝑃, there are elements of the lower and upper cut which are within that tolerance of each other.

Usually, proofs will be shorter if they use prarloc2 7466 instead. (Contributed by Jim Kingdon, 22-Oct-2019.)

Assertion
Ref Expression
prarloc ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑎𝐿𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃))
Distinct variable groups:   𝐿,𝑎,𝑏   𝑃,𝑎,𝑏   𝑈,𝑎,𝑏

Proof of Theorem prarloc
Dummy variables 𝑚 𝑛 𝑞 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prml 7439 . . . . . . 7 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥Q 𝑥𝐿)
2 df-rex 2454 . . . . . . 7 (∃𝑥Q 𝑥𝐿 ↔ ∃𝑥(𝑥Q𝑥𝐿))
31, 2sylib 121 . . . . . 6 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥(𝑥Q𝑥𝐿))
43adantr 274 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑥(𝑥Q𝑥𝐿))
5 prmu 7440 . . . . . . 7 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑦Q 𝑦𝑈)
6 df-rex 2454 . . . . . . 7 (∃𝑦Q 𝑦𝑈 ↔ ∃𝑦(𝑦Q𝑦𝑈))
75, 6sylib 121 . . . . . 6 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑦(𝑦Q𝑦𝑈))
87adantr 274 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑦(𝑦Q𝑦𝑈))
9 subhalfnqq 7376 . . . . . . . . 9 (𝑃Q → ∃𝑞Q (𝑞 +Q 𝑞) <Q 𝑃)
109adantl 275 . . . . . . . 8 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑞Q (𝑞 +Q 𝑞) <Q 𝑃)
11 df-rex 2454 . . . . . . . 8 (∃𝑞Q (𝑞 +Q 𝑞) <Q 𝑃 ↔ ∃𝑞(𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))
1210, 11sylib 121 . . . . . . 7 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑞(𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))
1312ancli 321 . . . . . 6 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ ∃𝑞(𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)))
14 19.42v 1899 . . . . . 6 (∃𝑞((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)) ↔ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ ∃𝑞(𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)))
1513, 14sylibr 133 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑞((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)))
16 eeeanv 1926 . . . . 5 (∃𝑥𝑦𝑞((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ↔ (∃𝑥(𝑥Q𝑥𝐿) ∧ ∃𝑦(𝑦Q𝑦𝑈) ∧ ∃𝑞((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))))
174, 8, 15, 16syl3anbrc 1176 . . . 4 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑥𝑦𝑞((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))))
18 prarloclemarch2 7381 . . . . . . . . . . . . . 14 ((𝑦Q𝑥Q𝑞Q) → ∃𝑛N (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))
19 df-rex 2454 . . . . . . . . . . . . . 14 (∃𝑛N (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))) ↔ ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
2018, 19sylib 121 . . . . . . . . . . . . 13 ((𝑦Q𝑥Q𝑞Q) → ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
21203com12 1202 . . . . . . . . . . . 12 ((𝑥Q𝑦Q𝑞Q) → ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
22213adant1r 1226 . . . . . . . . . . 11 (((𝑥Q𝑥𝐿) ∧ 𝑦Q𝑞Q) → ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
23223adant2r 1228 . . . . . . . . . 10 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ 𝑞Q) → ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
24233adant3r 1230 . . . . . . . . 9 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)) → ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
25243adant3l 1229 . . . . . . . 8 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
2625ancli 321 . . . . . . 7 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))))
27 19.42v 1899 . . . . . . 7 (∃𝑛(((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) ↔ (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))))
2826, 27sylibr 133 . . . . . 6 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → ∃𝑛(((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))))
29282eximi 1594 . . . . 5 (∃𝑦𝑞((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → ∃𝑦𝑞𝑛(((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))))
3029eximi 1593 . . . 4 (∃𝑥𝑦𝑞((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → ∃𝑥𝑦𝑞𝑛(((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))))
31 simpl1l 1043 . . . . . . . . . 10 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑥Q)
32 simp3rl 1065 . . . . . . . . . . 11 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → 𝑞Q)
3332adantr 274 . . . . . . . . . 10 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑞Q)
34 simp3rr 1066 . . . . . . . . . . 11 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → (𝑞 +Q 𝑞) <Q 𝑃)
3534adantr 274 . . . . . . . . . 10 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → (𝑞 +Q 𝑞) <Q 𝑃)
3631, 33, 353jca 1172 . . . . . . . . 9 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → (𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))
37 simp3ll 1063 . . . . . . . . . . . 12 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → ⟨𝐿, 𝑈⟩ ∈ P)
3837adantr 274 . . . . . . . . . . 11 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ⟨𝐿, 𝑈⟩ ∈ P)
39 simpl1r 1044 . . . . . . . . . . 11 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑥𝐿)
40 simprl 526 . . . . . . . . . . 11 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑛N)
41 simprrl 534 . . . . . . . . . . 11 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 1o <N 𝑛)
42 simprrr 535 . . . . . . . . . . . 12 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))
43 simpl2r 1046 . . . . . . . . . . . . 13 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑦𝑈)
44 prcunqu 7447 . . . . . . . . . . . . 13 ((⟨𝐿, 𝑈⟩ ∈ P𝑦𝑈) → (𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)) → (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))
4538, 43, 44syl2anc 409 . . . . . . . . . . . 12 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → (𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)) → (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))
4642, 45mpd 13 . . . . . . . . . . 11 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)
47 prarloclem 7463 . . . . . . . . . . 11 (((⟨𝐿, 𝑈⟩ ∈ P𝑥𝐿) ∧ (𝑛N𝑞Q ∧ 1o <N 𝑛) ∧ (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈) → ∃𝑚 ∈ ω ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))
4838, 39, 40, 33, 41, 46, 47syl231anc 1253 . . . . . . . . . 10 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ∃𝑚 ∈ ω ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))
49 df-rex 2454 . . . . . . . . . 10 (∃𝑚 ∈ ω ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈) ↔ ∃𝑚(𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))
5048, 49sylib 121 . . . . . . . . 9 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ∃𝑚(𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))
5136, 50jca 304 . . . . . . . 8 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ ∃𝑚(𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))
52 19.42v 1899 . . . . . . . 8 (∃𝑚((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))) ↔ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ ∃𝑚(𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))
5351, 52sylibr 133 . . . . . . 7 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ∃𝑚((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))
54 simprrl 534 . . . . . . . . . . . 12 (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))) → (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿)
55 eleq1 2233 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) → (𝑎𝐿 ↔ (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿))
5655anbi1d 462 . . . . . . . . . . . . . . . 16 (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) → ((𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈) ↔ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))
5756anbi2d 461 . . . . . . . . . . . . . . 15 (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) → ((𝑚 ∈ ω ∧ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)) ↔ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))
5857anbi2d 461 . . . . . . . . . . . . . 14 (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) → (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))) ↔ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))))
5958ceqsexgv 2859 . . . . . . . . . . . . 13 ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 → (∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))) ↔ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))))
6059biimprcd 159 . . . . . . . . . . . 12 (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))) → ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 → ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))))
6154, 60mpd 13 . . . . . . . . . . 11 (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))) → ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))))
62 simprrr 535 . . . . . . . . . . 11 (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))) → (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)
63 eleq1 2233 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → (𝑏𝑈 ↔ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))
6463anbi2d 461 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → ((𝑎𝐿𝑏𝑈) ↔ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))
6564anbi2d 461 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → ((𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)) ↔ (𝑚 ∈ ω ∧ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))
6665anbi2d 461 . . . . . . . . . . . . . . 15 (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))) ↔ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))))
6766anbi2d 461 . . . . . . . . . . . . . 14 (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → ((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) ↔ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))))
6867exbidv 1818 . . . . . . . . . . . . 13 (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → (∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) ↔ ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))))
6968ceqsexgv 2859 . . . . . . . . . . . 12 ((𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈 → (∃𝑏(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))))) ↔ ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))))
7069biimprcd 159 . . . . . . . . . . 11 (∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))) → ((𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈 → ∃𝑏(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))))))
7161, 62, 70sylc 62 . . . . . . . . . 10 (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))) → ∃𝑏(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))))))
72 19.42v 1899 . . . . . . . . . . 11 (∃𝑎(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))))) ↔ (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))))))
7372exbii 1598 . . . . . . . . . 10 (∃𝑏𝑎(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))))) ↔ ∃𝑏(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))))))
7471, 73sylibr 133 . . . . . . . . 9 (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))) → ∃𝑏𝑎(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))))))
75 simprrl 534 . . . . . . . . . . . . . 14 (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))) → 𝑎𝐿)
7675adantl 275 . . . . . . . . . . . . 13 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → 𝑎𝐿)
77 simprrr 535 . . . . . . . . . . . . . . 15 (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))) → 𝑏𝑈)
7877adantl 275 . . . . . . . . . . . . . 14 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → 𝑏𝑈)
79 simpl 108 . . . . . . . . . . . . . . 15 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))))
80 simprl2 1038 . . . . . . . . . . . . . . . 16 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → 𝑞Q)
81 simprl3 1039 . . . . . . . . . . . . . . . 16 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → (𝑞 +Q 𝑞) <Q 𝑃)
8280, 81jca 304 . . . . . . . . . . . . . . 15 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))
83 simprl1 1037 . . . . . . . . . . . . . . . 16 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → 𝑥Q)
84 simprrl 534 . . . . . . . . . . . . . . . 16 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → 𝑚 ∈ ω)
8583, 84jca 304 . . . . . . . . . . . . . . 15 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → (𝑥Q𝑚 ∈ ω))
86 prarloclemcalc 7464 . . . . . . . . . . . . . . 15 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑥Q𝑚 ∈ ω))) → 𝑏 <Q (𝑎 +Q 𝑃))
8779, 82, 85, 86syl12anc 1231 . . . . . . . . . . . . . 14 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → 𝑏 <Q (𝑎 +Q 𝑃))
8878, 87jca 304 . . . . . . . . . . . . 13 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃)))
8976, 88jca 304 . . . . . . . . . . . 12 (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → (𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
9089ancom1s 564 . . . . . . . . . . 11 (((𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ 𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞))) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈)))) → (𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
9190anasss 397 . . . . . . . . . 10 ((𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))))) → (𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
92912eximi 1594 . . . . . . . . 9 (∃𝑏𝑎(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))))) → ∃𝑏𝑎(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
9374, 92syl 14 . . . . . . . 8 (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))) → ∃𝑏𝑎(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
9493exlimiv 1591 . . . . . . 7 (∃𝑚((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))) → ∃𝑏𝑎(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
9553, 94syl 14 . . . . . 6 ((((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ∃𝑏𝑎(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
9695exlimivv 1889 . . . . 5 (∃𝑞𝑛(((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ∃𝑏𝑎(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
9796exlimivv 1889 . . . 4 (∃𝑥𝑦𝑞𝑛(((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ∃𝑏𝑎(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
9817, 30, 973syl 17 . . 3 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑏𝑎(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
99 excom 1657 . . 3 (∃𝑏𝑎(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))) ↔ ∃𝑎𝑏(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
10098, 99sylib 121 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑎𝑏(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
101 19.42v 1899 . . . . 5 (∃𝑏(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))) ↔ (𝑎𝐿 ∧ ∃𝑏(𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
102 df-rex 2454 . . . . . 6 (∃𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃) ↔ ∃𝑏(𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃)))
103102anbi2i 454 . . . . 5 ((𝑎𝐿 ∧ ∃𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃)) ↔ (𝑎𝐿 ∧ ∃𝑏(𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
104101, 103bitr4i 186 . . . 4 (∃𝑏(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))) ↔ (𝑎𝐿 ∧ ∃𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
105104exbii 1598 . . 3 (∃𝑎𝑏(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))) ↔ ∃𝑎(𝑎𝐿 ∧ ∃𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
106 df-rex 2454 . . 3 (∃𝑎𝐿𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃) ↔ ∃𝑎(𝑎𝐿 ∧ ∃𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
107105, 106bitr4i 186 . 2 (∃𝑎𝑏(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))) ↔ ∃𝑎𝐿𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃))
108100, 107sylib 121 1 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑎𝐿𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973   = wceq 1348  wex 1485  wcel 2141  wrex 2449  cop 3586   class class class wbr 3989  ωcom 4574  (class class class)co 5853  1oc1o 6388  2oc2o 6389   +o coa 6392  [cec 6511  Ncnpi 7234   <N clti 7237   ~Q ceq 7241  Qcnq 7242   +Q cplq 7244   ·Q cmq 7245   <Q cltq 7247   ~Q0 ceq0 7248   +Q0 cplq0 7251   ·Q0 cmq0 7252  Pcnp 7253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428
This theorem is referenced by:  prarloc2  7466  addlocpr  7498  prmuloc  7528  ltaddpr  7559  ltexprlemloc  7569  ltexprlemrl  7572  ltexprlemru  7574
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