Theorem caucvgprprlemlol 7758

Description: Lemma for caucvgprpr 7772. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)

Hypotheses

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

Assertion

Ref	Expression
caucvgprprlemlol	⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))

Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐹,𝑙   𝑢,𝐹,𝑟   𝑝,𝑙,𝑠   𝑞,𝑙,𝑠,𝑟   𝑡,𝑙,𝑝   𝑢,𝑞,𝑠,𝑟   𝑢,𝑝,𝑡,𝑟   𝜑,𝑟   𝑟,𝑞,𝑡

Allowed substitution hints:   𝜑(𝑢,𝑡,𝑘,𝑚,𝑛,𝑠,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑡,𝑘,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑡,𝑘,𝑛,𝑠,𝑞,𝑝)   𝐿(𝑢,𝑡,𝑘,𝑚,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemlol

Dummy variables 𝑏 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 7425	. . . . 5 ⊢ <Q ⊆ (Q × Q)
2	1	brel 4711	. . . 4 ⊢ (𝑠 <Q 𝑡 → (𝑠 ∈ Q ∧ 𝑡 ∈ Q))
3	2	simpld 112	. . 3 ⊢ (𝑠 <Q 𝑡 → 𝑠 ∈ Q)
4	3	3ad2ant2 1021	. 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑠 ∈ Q)
5		caucvgprpr.lim	. . . . . . 7 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩
6	5	caucvgprprlemell 7745	. . . . . 6 ⊢ (𝑡 ∈ (1st ‘𝐿) ↔ (𝑡 ∈ Q ∧ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
7	6	simprbi 275	. . . . 5 ⊢ (𝑡 ∈ (1st ‘𝐿) → ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏))
8	7	3ad2ant3 1022	. . . 4 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏))
9		simpll2 1039	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) → 𝑠 <Q 𝑡)
10		ltanqg 7460	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ (𝑧 +Q 𝑥) <Q (𝑧 +Q 𝑦)))
11	10	adantl 277	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q)) → (𝑥 <Q 𝑦 ↔ (𝑧 +Q 𝑥) <Q (𝑧 +Q 𝑦)))
12	4	ad2antrr 488	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) → 𝑠 ∈ Q)
13	2	simprd 114	. . . . . . . . . . . 12 ⊢ (𝑠 <Q 𝑡 → 𝑡 ∈ Q)
14	13	3ad2ant2 1021	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑡 ∈ Q)
15	14	ad2antrr 488	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) → 𝑡 ∈ Q)
16		simplr 528	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) → 𝑏 ∈ N)
17		nnnq 7482	. . . . . . . . . . 11 ⊢ (𝑏 ∈ N → [⟨𝑏, 1o⟩] ~Q ∈ Q)
18		recclnq 7452	. . . . . . . . . . 11 ⊢ ([⟨𝑏, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
19	16, 17, 18	3syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
20		addcomnqg 7441	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥))
21	20	adantl 277	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥))
22	11, 12, 15, 19, 21	caovord2d 6088	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) → (𝑠 <Q 𝑡 ↔ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))))
23	9, 22	mpbid 147	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
24		ltnqpri 7654	. . . . . . . 8 ⊢ ((𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) → ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩)
25	23, 24	syl 14	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) → ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩)
26		ltsopr 7656	. . . . . . . 8 ⊢ <P Or P
27		ltrelpr 7565	. . . . . . . 8 ⊢ <P ⊆ (P × P)
28	26, 27	sotri 5061	. . . . . . 7 ⊢ ((⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩ ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) → ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏))
29	25, 28	sylancom 420	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) → ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏))
30	29	ex 115	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) ∧ 𝑏 ∈ N) → (⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏) → ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
31	30	reximdva 2596	. . . 4 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → (∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏) → ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
32	8, 31	mpd 13	. . 3 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏))
33		opeq1 3804	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑟 → ⟨𝑏, 1o⟩ = ⟨𝑟, 1o⟩)
34	33	eceq1d 6623	. . . . . . . . . 10 ⊢ (𝑏 = 𝑟 → [⟨𝑏, 1o⟩] ~Q = [⟨𝑟, 1o⟩] ~Q )
35	34	fveq2d 5558	. . . . . . . . 9 ⊢ (𝑏 = 𝑟 → (*Q‘[⟨𝑏, 1o⟩] ~Q ) = (*Q‘[⟨𝑟, 1o⟩] ~Q ))
36	35	oveq2d 5934	. . . . . . . 8 ⊢ (𝑏 = 𝑟 → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
37	36	breq2d 4041	. . . . . . 7 ⊢ (𝑏 = 𝑟 → (𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))))
38	37	abbidv 2311	. . . . . 6 ⊢ (𝑏 = 𝑟 → {𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))})
39	36	breq1d 4039	. . . . . . 7 ⊢ (𝑏 = 𝑟 → ((𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞))
40	39	abbidv 2311	. . . . . 6 ⊢ (𝑏 = 𝑟 → {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞})
41	38, 40	opeq12d 3812	. . . . 5 ⊢ (𝑏 = 𝑟 → ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩)
42		fveq2 5554	. . . . 5 ⊢ (𝑏 = 𝑟 → (𝐹‘𝑏) = (𝐹‘𝑟))
43	41, 42	breq12d 4042	. . . 4 ⊢ (𝑏 = 𝑟 → (⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
44	43	cbvrexv 2727	. . 3 ⊢ (∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏) ↔ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟))
45	32, 44	sylib 122	. 2 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟))
46	5	caucvgprprlemell 7745	. 2 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
47	4, 45, 46	sylanbrc 417	1 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  {crab 2476  ⟨cop 3621   class class class wbr 4029  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918  1^st c1st 6191  1_oc1o 6462  [cec 6585  Ncnpi 7332   <_N clti 7335   ~_Q ceq 7339  Qcnq 7340   +_Q cplq 7342  *_Qcrq 7344   <_Q cltq 7345  Pcnp 7351   +_P cpp 7353  <_P cltp 7355

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-inp 7526  df-iltp 7530

This theorem is referenced by:  caucvgprprlemrnd  7761