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Theorem prarloc 7275
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 7276 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 7249 . . . . . . 7 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥Q 𝑥𝐿)
2 df-rex 2397 . . . . . . 7 (∃𝑥Q 𝑥𝐿 ↔ ∃𝑥(𝑥Q𝑥𝐿))
31, 2sylib 121 . . . . . 6 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥(𝑥Q𝑥𝐿))
43adantr 272 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑥(𝑥Q𝑥𝐿))
5 prmu 7250 . . . . . . 7 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑦Q 𝑦𝑈)
6 df-rex 2397 . . . . . . 7 (∃𝑦Q 𝑦𝑈 ↔ ∃𝑦(𝑦Q𝑦𝑈))
75, 6sylib 121 . . . . . 6 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑦(𝑦Q𝑦𝑈))
87adantr 272 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑦(𝑦Q𝑦𝑈))
9 subhalfnqq 7186 . . . . . . . . 9 (𝑃Q → ∃𝑞Q (𝑞 +Q 𝑞) <Q 𝑃)
109adantl 273 . . . . . . . 8 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑞Q (𝑞 +Q 𝑞) <Q 𝑃)
11 df-rex 2397 . . . . . . . 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 319 . . . . . 6 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ ∃𝑞(𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)))
14 19.42v 1860 . . . . . 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 1883 . . . . 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 1148 . . . 4 ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) → ∃𝑥𝑦𝑞((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))))
18 prarloclemarch2 7191 . . . . . . . . . . . . . 14 ((𝑦Q𝑥Q𝑞Q) → ∃𝑛N (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))
19 df-rex 2397 . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . 12 ((𝑥Q𝑦Q𝑞Q) → ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
22213adant1r 1192 . . . . . . . . . . 11 (((𝑥Q𝑥𝐿) ∧ 𝑦Q𝑞Q) → ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
23223adant2r 1194 . . . . . . . . . 10 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ 𝑞Q) → ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
24233adant3r 1196 . . . . . . . . 9 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)) → ∃𝑛(𝑛N ∧ (1o <N 𝑛𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
25243adant3l 1195 . . . . . . . 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 319 . . . . . . 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 1860 . . . . . . 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 1563 . . . . 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 1562 . . . 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 1015 . . . . . . . . . 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 1037 . . . . . . . . . . 11 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → 𝑞Q)
3332adantr 272 . . . . . . . . . 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 1038 . . . . . . . . . . 11 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → (𝑞 +Q 𝑞) <Q 𝑃)
3534adantr 272 . . . . . . . . . 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 1144 . . . . . . . . 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 1035 . . . . . . . . . . . 12 (((𝑥Q𝑥𝐿) ∧ (𝑦Q𝑦𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P𝑃Q) ∧ (𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → ⟨𝐿, 𝑈⟩ ∈ P)
3837adantr 272 . . . . . . . . . . 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 1016 . . . . . . . . . . 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 503 . . . . . . . . . . 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 511 . . . . . . . . . . 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 512 . . . . . . . . . . . 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 1018 . . . . . . . . . . . . 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 7257 . . . . . . . . . . . . 13 ((⟨𝐿, 𝑈⟩ ∈ P𝑦𝑈) → (𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)) → (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))
4538, 43, 44syl2anc 406 . . . . . . . . . . . 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 7273 . . . . . . . . . . 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 1219 . . . . . . . . . 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 2397 . . . . . . . . . 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 302 . . . . . . . 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 1860 . . . . . . . 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 511 . . . . . . . . . . . 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 2178 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) → (𝑎𝐿 ↔ (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿))
5655anbi1d 458 . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . 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 512 . . . . . . . . . . 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 2178 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → (𝑏𝑈 ↔ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))
6463anbi2d 457 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → ((𝑎𝐿𝑏𝑈) ↔ (𝑎𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))
6564anbi2d 457 . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 1779 . . . . . . . . . . . . 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 2786 . . . . . . . . . . . 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 1860 . . . . . . . . . . 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 1567 . . . . . . . . . 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 511 . . . . . . . . . . . . . 14 (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))) → 𝑎𝐿)
7675adantl 273 . . . . . . . . . . . . 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 512 . . . . . . . . . . . . . . 15 (((𝑥Q𝑞Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎𝐿𝑏𝑈))) → 𝑏𝑈)
7877adantl 273 . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . . . . 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 302 . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . . . . 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 302 . . . . . . . . . . . . . . 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 7274 . . . . . . . . . . . . . . 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 1197 . . . . . . . . . . . . . 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 302 . . . . . . . . . . . . 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 302 . . . . . . . . . . . 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 541 . . . . . . . . . . 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 394 . . . . . . . . . 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 1563 . . . . . . . . 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 1560 . . . . . . 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 1850 . . . . 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 1850 . . . 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 1625 . . 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 1860 . . . . 5 (∃𝑏(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))) ↔ (𝑎𝐿 ∧ ∃𝑏(𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))))
102 df-rex 2397 . . . . . 6 (∃𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃) ↔ ∃𝑏(𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃)))
103102anbi2i 450 . . . . 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 1567 . . 3 (∃𝑎𝑏(𝑎𝐿 ∧ (𝑏𝑈𝑏 <Q (𝑎 +Q 𝑃))) ↔ ∃𝑎(𝑎𝐿 ∧ ∃𝑏𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
106 df-rex 2397 . . 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 945   = wceq 1314  wex 1451  wcel 1463  wrex 2392  cop 3498   class class class wbr 3897  ωcom 4472  (class class class)co 5740  1oc1o 6272  2oc2o 6273   +o coa 6276  [cec 6393  Ncnpi 7044   <N clti 7047   ~Q ceq 7051  Qcnq 7052   +Q cplq 7054   ·Q cmq 7055   <Q cltq 7057   ~Q0 ceq0 7058   +Q0 cplq0 7061   ·Q0 cmq0 7062  Pcnp 7063
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238
This theorem is referenced by:  prarloc2  7276  addlocpr  7308  prmuloc  7338  ltaddpr  7369  ltexprlemloc  7379  ltexprlemrl  7382  ltexprlemru  7384
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