Theorem caucvgprlemloc 6865

Description: Lemma for caucvgpr 6872. 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 6599	. . . . 5 ⊢ (𝑠 <Q 𝑟 → ∃𝑦 ∈ Q (𝑠 +Q 𝑦) = 𝑟)
2	1	adantl 271	. . . 4 ⊢ (((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) → ∃𝑦 ∈ Q (𝑠 +Q 𝑦) = 𝑟)
3		subhalfnqq 6604	. . . . . 6 ⊢ (𝑦 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝑦)
4	3	ad2antrl 473	. . . . 5 ⊢ ((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝑦)
5		archrecnq 6853	. . . . . . 7 ⊢ (𝑥 ∈ Q → ∃𝑚 ∈ N (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) <Q 𝑥)
6	5	ad2antrl 473	. . . . . 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 498	. . . . . . . . . . . 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 6612	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ N → [⟨𝑚, 1𝑜⟩] ~Q ∈ Q)
9		recclnq 6582	. . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . 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 501	. . . . . . . . . . . . 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 6594	. . . . . . . . . . . . 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 1170	. . . . . . . . . . . 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 423	. . . . . . . . . . 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 502	. . . . . . . . . . 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 6588	. . . . . . . . . . . 12 ⊢ <Q Or Q
18		ltrelnq 6555	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
19	17, 18	sotri 4740	. . . . . . . . . . 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 403	. . . . . . . . . 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 501	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) → 𝑠 ∈ Q)
22	21	ad3antrrr 475	. . . . . . . . . 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 6592	. . . . . . . . . 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 403	. . . . . . . . 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 498	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) ∧ (𝑦 ∈ Q ∧ (𝑠 +Q 𝑦) = 𝑟)) → (𝑠 +Q 𝑦) = 𝑟)
26	25	ad2antrr 471	. . . . . . . . 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 3809	. . . . . . . 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 6565	. . . . . . . . . . 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 403	. . . . . . . . . 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 6565	. . . . . . . . . 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 403	. . . . . . . . 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 502	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) → 𝑟 ∈ Q)
33	32	ad3antrrr 475	. . . . . . . . 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 479	. . . . . . . . . . 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 497	. . . . . . . . . . 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	ffvelrnd 5324	. . . . . . . . . 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 6565	. . . . . . . . . 10 ⊢ (((𝐹‘𝑚) ∈ Q ∧ (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q) → ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) ∈ Q)
39	37, 11, 38	syl2anc 403	. . . . . . . . 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 4075	. . . . . . . . . 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 414	. . . . . . . . 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 1169	. . . . . . . 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 270	. . . . . . . . . 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 501	. . . . . . . . . . 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 108	. . . . . . . . . . . . 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 270	. . . . . . . . . . . . . . 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 6572	. . . . . . . . . . . . . . 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 1169	. . . . . . . . . . . . . 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 3795	. . . . . . . . . . . . 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 165	. . . . . . . . . . . 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 6590	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
53	52	adantl 271	. . . . . . . . . . . . 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 6565	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ Q ∧ (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) ∈ Q)
55	44, 47, 54	syl2anc 403	. . . . . . . . . . . . 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 270	. . . . . . . . . . . . 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 6571	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
58	57	adantl 271	. . . . . . . . . . . . 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 5690	. . . . . . . . . . . 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 165	. . . . . . . . . . 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 3570	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → ⟨𝑗, 1𝑜⟩ = ⟨𝑚, 1𝑜⟩)
62	61	eceq1d 6165	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑚, 1𝑜⟩] ~Q )
63	62	fveq2d 5202	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑚 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑚, 1𝑜⟩] ~Q ))
64	63	oveq2d 5548	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )))
65		fveq2 5198	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (𝐹‘𝑗) = (𝐹‘𝑚))
66	64, 65	breq12d 3798	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑚 → ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑚)))
67	66	rspcev 2701	. . . . . . . . . . 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 403	. . . . . . . . . 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 5539	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
70	69	breq1d 3795	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)))
71	70	rexbidv 2369	. . . . . . . . . . 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 5201	. . . . . . . . . . . 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 6553	. . . . . . . . . . . . . 14 ⊢ Q ∈ V
75	74	rabex 3922	. . . . . . . . . . . . 13 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
76	74	rabex 3922	. . . . . . . . . . . . 13 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
77	75, 76	op1st 5793	. . . . . . . . . . . 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 2101	. . . . . . . . . . 11 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}
79	71, 78	elrab2 2751	. . . . . . . . . 10 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)))
80	44, 68, 79	sylanbrc 408	. . . . . . . . 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 113	. . . . . . . 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 270	. . . . . . . . . 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 5550	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )))
84	83	breq1d 3795	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑚 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟 ↔ ((𝐹‘𝑚) +Q (*Q‘[⟨𝑚, 1𝑜⟩] ~Q )) <Q 𝑟))
85	84	rspcev 2701	. . . . . . . . . . 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 277	. . . . . . . . . 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 3789	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑟 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
88	87	rexbidv 2369	. . . . . . . . . . 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 5201	. . . . . . . . . . . 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 5794	. . . . . . . . . . . 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 2101	. . . . . . . . . . 11 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
92	88, 91	elrab2 2751	. . . . . . . . . 10 ⊢ (𝑟 ∈ (2nd ‘𝐿) ↔ (𝑟 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑟))
93	82, 86, 92	sylanbrc 408	. . . . . . . . 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 113	. . . . . . . 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 732	. . . . . . 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 2481	. . . . 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 2481	. . . 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 2481	. . 3 ⊢ (((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) ∧ 𝑠 <Q 𝑟) → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿)))
100	99	ex 113	. 2 ⊢ ((𝜑 ∧ (𝑠 ∈ Q ∧ 𝑟 ∈ Q)) → (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))
101	100	ralrimivva 2443	1 ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349  {crab 2352  ⟨cop 3401   class class class wbr 3785   Or wor 4050  ⟶wf 4918  ‘cfv 4922  (class class class)co 5532  1^st c1st 5785  2^nd c2nd 5786  1_𝑜c1o 6017  [cec 6127  Ncnpi 6462   <_N clti 6465   ~_Q ceq 6469  Qcnq 6470   +_Q cplq 6472  *_Qcrq 6474   <_Q cltq 6475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543

This theorem is referenced by:  caucvgprlemcl  6866