Theorem prarloc 7516

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 7517 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 7490	. . . . . . 7 ⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ 𝐿)
2		df-rex 2471	. . . . . . 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 7491	. . . . . . 7 ⊢ (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑦 ∈ Q 𝑦 ∈ 𝑈)
6		df-rex 2471	. . . . . . 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 7427	. . . . . . . . 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 2471	. . . . . . . 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 1916	. . . . . 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 1943	. . . . 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 1182	. . . 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 7432	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Q ∧ 𝑥 ∈ Q ∧ 𝑞 ∈ Q) → ∃𝑛 ∈ N (1o <N 𝑛 ∧ 𝑦 <Q (𝑥 +Q ([⟨𝑛, 1o⟩] ~Q ·Q 𝑞))))
19		df-rex 2471	. . . . . . . . . . . . . 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 1208	. . . . . . . . . . . 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 1232	. . . . . . . . . . 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 1234	. . . . . . . . . 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 1236	. . . . . . . . 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 1235	. . . . . . . 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 1916	. . . . . . 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 1611	. . . . 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 1610	. . . 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 1049	. . . . . . . . . 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 1071	. . . . . . . . . . 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 1072	. . . . . . . . . . 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 1178	. . . . . . . . 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 1069	. . . . . . . . . . . 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 1050	. . . . . . . . . . 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 1052	. . . . . . . . . . . . 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 7498	. . . . . . . . . . . . 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 7514	. . . . . . . . . . 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 1268	. . . . . . . . . 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 2471	. . . . . . . . . 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 1916	. . . . . . . 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 2250	. . . . . . . . . . . . . . . . 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 2878	. . . . . . . . . . . . 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 2250	. . . . . . . . . . . . . . . . . 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 1835	. . . . . . . . . . . . 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 2878	. . . . . . . . . . . 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 1916	. . . . . . . . . . 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 1615	. . . . . . . . . 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 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)
81		simprl3 1045	. . . . . . . . . . . . . . . 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 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)))) → 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 7515	. . . . . . . . . . . . . . 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 1246	. . . . . . . . . . . . . 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 1611	. . . . . . . . 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 1608	. . . . . . 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 1906	. . . . 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 1906	. . . 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 1674	. . 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 1916	. . . . 5 ⊢ (∃𝑏(𝑎 ∈ 𝐿 ∧ (𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))) ↔ (𝑎 ∈ 𝐿 ∧ ∃𝑏(𝑏 ∈ 𝑈 ∧ 𝑏 <Q (𝑎 +Q 𝑃))))
102		df-rex 2471	. . . . . 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 1615	. . 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 2471	. . 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 979   = wceq 1363  ∃wex 1502   ∈ wcel 2158  ∃wrex 2466  ⟨cop 3607   class class class wbr 4015  ωcom 4601  (class class class)co 5888  1_oc1o 6424  2_oc2o 6425   +_o coa 6428  [cec 6547  Ncnpi 7285   <_N clti 7288   ~_Q ceq 7292  Qcnq 7293   +_Q cplq 7295   ·_Q cmq 7296   <_Q cltq 7298   ~_Q0 ceq0 7299   +_Q0 cplq0 7302   ·_Q0 cmq0 7303  Pcnp 7304

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-1o 6431  df-2o 6432  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-pli 7318  df-mi 7319  df-lti 7320  df-plpq 7357  df-mpq 7358  df-enq 7360  df-nqqs 7361  df-plqqs 7362  df-mqqs 7363  df-1nqqs 7364  df-rq 7365  df-ltnqqs 7366  df-enq0 7437  df-nq0 7438  df-0nq0 7439  df-plq0 7440  df-mq0 7441  df-inp 7479

This theorem is referenced by:  prarloc2  7517  addlocpr  7549  prmuloc  7579  ltaddpr  7610  ltexprlemloc  7620  ltexprlemrl  7623  ltexprlemru  7625