Theorem caucvgprlemloc 7649

Description: Lemma for caucvgpr 7656. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	⊢ (𝜑 → 𝐹:N⟶Q)
caucvgpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd	⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
caucvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩

Assertion

Ref	Expression
caucvgprlemloc	⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))

Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑙   𝑢,𝐹   𝜑,𝑗,𝑟,𝑠   𝑠,𝑙   𝑢,𝑗,𝑟

Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑠,𝑟,𝑙)   𝐹(𝑘,𝑛,𝑠,𝑟)   𝐿(𝑢,𝑗,𝑘,𝑛,𝑠,𝑟,𝑙)

Proof of Theorem caucvgprlemloc

Dummy variables 𝑓 𝑔 ℎ 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexnqi 7383	. . . . 5 ⊢ (𝑠 <Q 𝑟 → ∃𝑦 ∈ Q (𝑠 +Q 𝑦) = 𝑟)
2	1	adantl 277	. . . 4 ⊢ (((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) → ∃𝑦 ∈ Q (𝑠 +Q 𝑦) = 𝑟)
3		subhalfnqq 7388	. . . . . 6 ⊢ (𝑦 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝑦)
4	3	ad2antrl 490	. . . . 5 ⊢ ((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝑦)
5		archrecnq 7637	. . . . . . 7 ⊢ (𝑥 ∈ Q → ∃𝑚 ∈ N (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)
6	5	ad2antrl 490	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → ∃𝑚 ∈ N (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)
7		simprr 531	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)
8		nnnq 7396	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ N → [⟨𝑚, 1o⟩] ~Q ∈ Q)
9		recclnq 7366	. . . . . . . . . . . . . . 15 ⊢ ([⟨𝑚, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q)
10	8, 9	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ N → (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q)
11	10	ad2antrl 490	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q)
12		simplrl 535	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → 𝑥 ∈ Q)
13		lt2addnq 7378	. . . . . . . . . . . . 13 ⊢ ((((*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q ∧ 𝑥 ∈ Q) ∧ ((*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q ∧ 𝑥 ∈ Q)) → (((*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥 ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝑥 +Q 𝑥)))
14	11, 12, 11, 12, 13	syl22anc 1239	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (((*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥 ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝑥 +Q 𝑥)))
15	7, 7, 14	mp2and 433	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝑥 +Q 𝑥))
16		simplrr 536	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑥 +Q 𝑥) <Q 𝑦)
17		ltsonq 7372	. . . . . . . . . . . 12 ⊢ <Q Or Q
18		ltrelnq 7339	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
19	17, 18	sotri 5016	. . . . . . . . . . 11 ⊢ ((((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝑥 +Q 𝑥) ∧ (𝑥 +Q 𝑥) <Q 𝑦) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑦)
20	15, 16, 19	syl2anc 411	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑦)
21		simplrl 535	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) → 𝑠 ∈ Q)
22	21	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → 𝑠 ∈ Q)
23		ltanqi 7376	. . . . . . . . . 10 ⊢ ((((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑦 ∧ 𝑠 ∈ Q) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q (𝑠 +Q 𝑦))
24	20, 22, 23	syl2anc 411	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q (𝑠 +Q 𝑦))
25		simprr 531	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) → (𝑠 +Q 𝑦) = 𝑟)
26	25	ad2antrr 488	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑠 +Q 𝑦) = 𝑟)
27	24, 26	breqtrd 4024	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q 𝑟)
28		addclnq 7349	. . . . . . . . . . 11 ⊢ (((*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
29	11, 11, 28	syl2anc 411	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
30		addclnq 7349	. . . . . . . . . 10 ⊢ ((𝑠 ∈ Q ∧ ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∈ Q)
31	22, 29, 30	syl2anc 411	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∈ Q)
32		simplrr 536	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) → 𝑟 ∈ Q)
33	32	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → 𝑟 ∈ Q)
34		caucvgpr.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:N⟶Q)
35	34	ad5antr 496	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → 𝐹:N⟶Q)
36		simprl 529	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → 𝑚 ∈ N)
37	35, 36	ffvelcdmd 5644	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝐹‘𝑚) ∈ Q)
38		addclnq 7349	. . . . . . . . . 10 ⊢ (((𝐹‘𝑚) ∈ Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q) → ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
39	37, 11, 38	syl2anc 411	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
40		sowlin 4314	. . . . . . . . . 10 ⊢ (( <Q Or Q ∧ ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∈ Q ∧ 𝑟 ∈ Q ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)) → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q 𝑟 → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∨ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟)))
41	17, 40	mpan 424	. . . . . . . . 9 ⊢ (((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∈ Q ∧ 𝑟 ∈ Q ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q) → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q 𝑟 → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∨ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟)))
42	31, 33, 39, 41	syl3anc 1238	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q 𝑟 → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∨ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟)))
43	27, 42	mpd 13	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∨ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟))
44	22	adantr 276	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → 𝑠 ∈ Q)
45		simplrl 535	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → 𝑚 ∈ N)
46		simpr 110	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )))
47	11	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q)
48		addassnqg 7356	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q) → ((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) = (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))))
49	44, 47, 47, 48	syl3anc 1238	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → ((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) = (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))))
50	49	breq1d 4008	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ↔ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))))
51	46, 50	mpbird 167	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → ((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )))
52		ltanqg 7374	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
53	52	adantl 277	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
54		addclnq 7349	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ Q ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
55	44, 47, 54	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∈ Q)
56	37	adantr 276	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (𝐹‘𝑚) ∈ Q)
57		addcomnqg 7355	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
58	57	adantl 277	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
59	53, 55, 56, 47, 58	caovord2d 6034	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → ((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝐹‘𝑚) ↔ ((𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))))
60	51, 59	mpbird 167	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝐹‘𝑚))
61		opeq1 3774	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → ⟨𝑗, 1o⟩ = ⟨𝑚, 1o⟩)
62	61	eceq1d 6561	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑚, 1o⟩] ~Q )
63	62	fveq2d 5511	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑚 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑚, 1o⟩] ~Q ))
64	63	oveq2d 5881	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )))
65		fveq2 5507	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (𝐹‘𝑗) = (𝐹‘𝑚))
66	64, 65	breq12d 4011	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑚 → ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝐹‘𝑚)))
67	66	rspcev 2839	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ N ∧ (𝑠 +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q (𝐹‘𝑚)) → ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
68	45, 60, 67	syl2anc 411	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
69		oveq1 5872	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
70	69	breq1d 4008	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
71	70	rexbidv 2476	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑠 → (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
72		caucvgpr.lim	. . . . . . . . . . . . 13 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
73	72	fveq2i 5510	. . . . . . . . . . . 12 ⊢ (1st ‘𝐿) = (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
74		nqex 7337	. . . . . . . . . . . . . 14 ⊢ Q ∈ V
75	74	rabex 4142	. . . . . . . . . . . . 13 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
76	74	rabex 4142	. . . . . . . . . . . . 13 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
77	75, 76	op1st 6137	. . . . . . . . . . . 12 ⊢ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
78	73, 77	eqtri 2196	. . . . . . . . . . 11 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
79	71, 78	elrab2 2894	. . . . . . . . . 10 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
80	44, 68, 79	sylanbrc 417	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ (𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) → 𝑠 ∈ (1st ‘𝐿))
81	80	ex 115	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → ((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) → 𝑠 ∈ (1st ‘𝐿)))
82	33	adantr 276	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟) → 𝑟 ∈ Q)
83	65, 63	oveq12d 5883	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )))
84	83	breq1d 4008	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑚 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟 ↔ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟))
85	84	rspcev 2839	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ N ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
86	36, 85	sylan 283	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
87		breq2 4002	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑟 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
88	87	rexbidv 2476	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑟 → (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
89	72	fveq2i 5510	. . . . . . . . . . . 12 ⊢ (2nd ‘𝐿) = (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
90	75, 76	op2nd 6138	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
91	89, 90	eqtri 2196	. . . . . . . . . . 11 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
92	88, 91	elrab2 2894	. . . . . . . . . 10 ⊢ (𝑟 ∈ (2nd ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
93	82, 86, 92	sylanbrc 417	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) ∧ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟) → 𝑟 ∈ (2nd ‘𝐿))
94	93	ex 115	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟 → 𝑟 ∈ (2nd ‘𝐿)))
95	81, 94	orim12d 786	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (((𝑠 +Q ((*Q‘[⟨𝑚, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q ))) <Q ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) ∨ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1o⟩] ~Q )) <Q 𝑟) → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))
96	43, 95	mpd 13	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) ∧ (𝑚 ∈ N ∧ (*Q‘[⟨𝑚, 1o⟩] ~Q ) <Q 𝑥)) → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿)))
97	6, 96	rexlimddv 2597	. . . . 5 ⊢ (((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝑦)) → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿)))
98	4, 97	rexlimddv 2597	. . . 4 ⊢ ((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿)))
99	2, 98	rexlimddv 2597	. . 3 ⊢ (((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿)))
100	99	ex 115	. 2 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) → (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))
101	100	ralrimivva 2557	1 ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2146  ∀wral 2453  ∃wrex 2454  {crab 2457  ⟨cop 3592   class class class wbr 3998   Or wor 4289  ⟶wf 5204  ‘cfv 5208  (class class class)co 5865  1^st c1st 6129  2^nd c2nd 6130  1_oc1o 6400  [cec 6523  Ncnpi 7246   <_N clti 7249   ~_Q ceq 7253  Qcnq 7254   +_Q cplq 7256  *_Qcrq 7258   <_Q cltq 7259

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581

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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327

This theorem is referenced by:  caucvgprlemcl  7650