Theorem prarloc 7505

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 7506 instead. (Contributed by Jim Kingdon, 22-Oct-2019.)

Assertion

Ref	Expression
prarloc	⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 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.

Step	Hyp	Ref	Expression
1		prml 7479	. . . . . . 7 ⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝐿)
2		df-rex 2461	. . . . . . 7 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ 𝐿 ↔ ∃𝑥(𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿))
3	1, 2	sylib 122	. . . . . 6 ⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥(𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿))
4	3	adantr 276	. . . . 5 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑥(𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿))
5		prmu 7480	. . . . . . 7 ⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑦 ∈ Q 𝑦 ∈ 𝑈)
6		df-rex 2461	. . . . . . 7 ⊢ (∃𝑦 ∈ Q 𝑦 ∈ 𝑈 ↔ ∃𝑦(𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈))
7	5, 6	sylib 122	. . . . . 6 ⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑦(𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈))
8	7	adantr 276	. . . . 5 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑦(𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈))
9		subhalfnqq 7416	. . . . . . . . 9 ⊢ (𝑃 ∈ Q → ∃𝑞 ∈ Q (𝑞 +Q 𝑞) <Q 𝑃)
10	9	adantl 277	. . . . . . . 8 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑞 ∈ Q (𝑞 +Q 𝑞) <Q 𝑃)
11		df-rex 2461	. . . . . . . 8 ⊢ (∃𝑞 ∈ Q (𝑞 +Q 𝑞) <Q 𝑃 ↔ ∃𝑞(𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))
12	10, 11	sylib 122	. . . . . . 7 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑞(𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))
13	12	ancli 323	. . . . . 6 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ ∃𝑞(𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)))
14		19.42v 1906	. . . . . 6 ⊢ (∃𝑞((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)) ↔ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ ∃𝑞(𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)))
15	13, 14	sylibr 134	. . . . 5 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑞((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)))
16		eeeanv 1933	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑞((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ↔ (∃𝑥(𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ ∃𝑦(𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ∃𝑞((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))))
17	4, 8, 15, 16	syl3anbrc 1181	. . . 4 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑥∃𝑦∃𝑞((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))))
18		prarloclemarch2 7421	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Q ∧ 𝑥 ∈ Q ∧ 𝑞 ∈ Q) → ∃𝑛 ∈ N (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))
19		df-rex 2461	. . . . . . . . . . . . . 14 ⊢ (∃𝑛 ∈ N (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))) ↔ ∃𝑛(𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
20	18, 19	sylib 122	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Q ∧ 𝑥 ∈ Q ∧ 𝑞 ∈ Q) → ∃𝑛(𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
21	20	3com12 1207	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑞 ∈ Q) → ∃𝑛(𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
22	21	3adant1r 1231	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ 𝑦 ∈ Q ∧ 𝑞 ∈ Q) → ∃𝑛(𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
23	22	3adant2r 1233	. . . . . . . . . 10 ⊢ (((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ 𝑞 ∈ Q) → ∃𝑛(𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
24	23	3adant3r 1235	. . . . . . . . 9 ⊢ (((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃)) → ∃𝑛(𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
25	24	3adant3l 1234	. . . . . . . 8 ⊢ (((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → ∃𝑛(𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))))
26	25	ancli 323	. . . . . . 7 ⊢ (((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → (((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ ∃𝑛(𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))))
27		19.42v 1906	. . . . . . 7 ⊢ (∃𝑛(((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) ↔ (((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ ∃𝑛(𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))))
28	26, 27	sylibr 134	. . . . . 6 ⊢ (((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → ∃𝑛(((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))))
29	28	2eximi 1601	. . . . 5 ⊢ (∃𝑦∃𝑞((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → ∃𝑦∃𝑞∃𝑛(((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))))
30	29	eximi 1600	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑞((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → ∃𝑥∃𝑦∃𝑞∃𝑛(((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))))
31		simpl1l 1048	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑥 ∈ Q)
32		simp3rl 1070	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → 𝑞 ∈ Q)
33	32	adantr 276	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑞 ∈ Q)
34		simp3rr 1071	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → (𝑞 +Q 𝑞) <Q 𝑃)
35	34	adantr 276	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → (𝑞 +Q 𝑞) <Q 𝑃)
36	31, 33, 35	3jca 1177	. . . . . . . . 9 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → (𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))
37		simp3ll 1068	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) → ⟨𝐿, 𝑈⟩ ∈ P)
38	37	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ⟨𝐿, 𝑈⟩ ∈ P)
39		simpl1r 1049	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑥 ∈ 𝐿)
40		simprl 529	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑛 ∈ N)
41		simprrl 539	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 1o <N 𝑛)
42		simprrr 540	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)))
43		simpl2r 1051	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → 𝑦 ∈ 𝑈)
44		prcunqu 7487	. . . . . . . . . . . . 13 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑦 ∈ 𝑈) → (𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)) → (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))
45	38, 43, 44	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <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))
46	42, 45	mpd 13	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)
47		prarloclem 7503	. . . . . . . . . . 11 ⊢ (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑥 ∈ 𝐿) ∧ (𝑛 ∈ N ∧ 𝑞 ∈ Q ∧ 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))
48	38, 39, 40, 33, 41, 46, 47	syl231anc 1258	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <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 2461	. . . . . . . . . 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)))
50	48, 49	sylib 122	. . . . . . . . 9 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <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)))
51	36, 50	jca 306	. . . . . . . 8 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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 1906	. . . . . . . 8 ⊢ (∃𝑚((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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 ∧ 𝑞 ∈ Q ∧ (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))))
53	51, 52	sylibr 134	. . . . . . 7 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ∃𝑚((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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 539	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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 2240	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) → (𝑎 ∈ 𝐿 ↔ (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿))
56	55	anbi1d 465	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) → ((𝑎 ∈ 𝐿 ∧ (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)))
57	56	anbi2d 464	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) → ((𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ (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))))
58	57	anbi2d 464	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) → (((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))) ↔ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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)))))
59	58	ceqsexgv 2868	. . . . . . . . . . . . 13 ⊢ ((𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∈ 𝐿 → (∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))) ↔ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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)))))
60	59	biimprcd 160	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))))
61	54, 60	mpd 13	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))))
62		simprrr 540	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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 2240	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → (𝑏 ∈ 𝑈 ↔ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))
64	63	anbi2d 464	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → ((𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈) ↔ (𝑎 ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))
65	64	anbi2d 464	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → ((𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)) ↔ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))
66	65	anbi2d 464	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → (((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))) ↔ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈)))))
67	66	anbi2d 464	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → ((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) ↔ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))))
68	67	exbidv 1825	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) → (∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) ↔ ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))))
69	68	ceqsexgv 2868	. . . . . . . . . . . 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)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))))) ↔ ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∈ 𝑈))))))
70	69	biimprcd 160	. . . . . . . . . . 11 ⊢ (∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ (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)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))))))
71	61, 62, 70	sylc 62	. . . . . . . . . 10 ⊢ (((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))))))
72		19.42v 1906	. . . . . . . . . . 11 ⊢ (∃𝑎(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))))) ↔ (𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))))))
73	72	exbii 1605	. . . . . . . . . 10 ⊢ (∃𝑏∃𝑎(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))))) ↔ ∃𝑏(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ ∃𝑎(𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))))))
74	71, 73	sylibr 134	. . . . . . . . 9 ⊢ (((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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∃𝑎(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))))))
75		simprrl 539	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))) → 𝑎 ∈ 𝐿)
76	75	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → 𝑎 ∈ 𝐿)
77		simprrr 540	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))) → 𝑏 ∈ 𝑈)
78	77	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → 𝑏 ∈ 𝑈)
79		simpl 109	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))))
80		simprl2 1043	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → 𝑞 ∈ Q)
81		simprl3 1044	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → (𝑞 +Q 𝑞) <Q 𝑃)
82	80, 81	jca 306	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))
83		simprl1 1042	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → 𝑥 ∈ Q)
84		simprrl 539	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → 𝑚 ∈ ω)
85	83, 84	jca 306	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → (𝑥 ∈ Q ∧ 𝑚 ∈ ω))
86		prarloclemcalc 7504	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑥 ∈ Q ∧ 𝑚 ∈ ω))) → 𝑏 <Q (𝑎 +Q 𝑃))
87	79, 82, 85, 86	syl12anc 1236	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → 𝑏 <Q (𝑎 +Q 𝑃))
88	78, 87	jca 306	. . . . . . . . . . . . 13 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃)))
89	76, 88	jca 306	. . . . . . . . . . . 12 ⊢ (((𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ 𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → (𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
90	89	ancom1s 569	. . . . . . . . . . 11 ⊢ (((𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ 𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞))) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈)))) → (𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
91	90	anasss 399	. . . . . . . . . 10 ⊢ ((𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))))) → (𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
92	91	2eximi 1601	. . . . . . . . 9 ⊢ (∃𝑏∃𝑎(𝑏 = (𝑥 +Q ([⟨(𝑚 +o 2o), 1o⟩] ~Q ·Q 𝑞)) ∧ (𝑎 = (𝑥 +Q0 ([⟨𝑚, 1o⟩] ~Q0 ·Q0 𝑞)) ∧ ((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃) ∧ (𝑚 ∈ ω ∧ (𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝑈))))) → ∃𝑏∃𝑎(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
93	74, 92	syl 14	. . . . . . . 8 ⊢ (((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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∃𝑎(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
94	93	exlimiv 1598	. . . . . . 7 ⊢ (∃𝑚((𝑥 ∈ Q ∧ 𝑞 ∈ Q ∧ (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∃𝑎(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
95	53, 94	syl 14	. . . . . 6 ⊢ ((((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ∃𝑏∃𝑎(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
96	95	exlimivv 1896	. . . . 5 ⊢ (∃𝑞∃𝑛(((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ∃𝑏∃𝑎(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
97	96	exlimivv 1896	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑞∃𝑛(((𝑥 ∈ Q ∧ 𝑥 ∈ 𝐿) ∧ (𝑦 ∈ Q ∧ 𝑦 ∈ 𝑈) ∧ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) ∧ (𝑞 ∈ Q ∧ (𝑞 +Q 𝑞) <Q 𝑃))) ∧ (𝑛 ∈ N ∧ (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))) → ∃𝑏∃𝑎(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
98	17, 30, 97	3syl 17	. . 3 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑏∃𝑎(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
99		excom 1664	. . 3 ⊢ (∃𝑏∃𝑎(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))) ↔ ∃𝑎∃𝑏(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
100	98, 99	sylib 122	. 2 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎∃𝑏(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
101		19.42v 1906	. . . . 5 ⊢ (∃𝑏(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))) ↔ (𝑎 ∈ 𝐿 ∧ ∃𝑏(𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
102		df-rex 2461	. . . . . 6 ⊢ (∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃) ↔ ∃𝑏(𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃)))
103	102	anbi2i 457	. . . . 5 ⊢ ((𝑎 ∈ 𝐿 ∧ ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃)) ↔ (𝑎 ∈ 𝐿 ∧ ∃𝑏(𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
104	101, 103	bitr4i 187	. . . 4 ⊢ (∃𝑏(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))) ↔ (𝑎 ∈ 𝐿 ∧ ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
105	104	exbii 1605	. . 3 ⊢ (∃𝑎∃𝑏(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))) ↔ ∃𝑎(𝑎 ∈ 𝐿 ∧ ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
106		df-rex 2461	. . 3 ⊢ (∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃) ↔ ∃𝑎(𝑎 ∈ 𝐿 ∧ ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
107	105, 106	bitr4i 187	. 2 ⊢ (∃𝑎∃𝑏(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))) ↔ ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃))
108	100, 107	sylib 122	1 ⊢ ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 ∃𝑏 ∈ 𝑈 𝑏 <Q (𝑎 +Q 𝑃))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∃wrex 2456  ⟨cop 3597   class class class wbr 4005  ωcom 4591  (class class class)co 5878  1_oc1o 6413  2_oc2o 6414   +_o coa 6417  [cec 6536  Ncnpi 7274   <_N clti 7277   ~_Q ceq 7281  Qcnq 7282   +_Q cplq 7284   ·_Q cmq 7285   <_Q cltq 7287   ~_Q0 ceq0 7288   +_Q0 cplq0 7291   ·_Q0 cmq0 7292  Pcnp 7293

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-1o 6420  df-2o 6421  df-oadd 6424  df-omul 6425  df-er 6538  df-ec 6540  df-qs 6544  df-ni 7306  df-pli 7307  df-mi 7308  df-lti 7309  df-plpq 7346  df-mpq 7347  df-enq 7349  df-nqqs 7350  df-plqqs 7351  df-mqqs 7352  df-1nqqs 7353  df-rq 7354  df-ltnqqs 7355  df-enq0 7426  df-nq0 7427  df-0nq0 7428  df-plq0 7429  df-mq0 7430  df-inp 7468

This theorem is referenced by:  prarloc2  7506  addlocpr  7538  prmuloc  7568  ltaddpr  7599  ltexprlemloc  7609  ltexprlemrl  7612  ltexprlemru  7614