Theorem caucvgprprlemlol 8065

Description: Lemma for caucvgprpr 8079. 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 7732	. . . . 5 ⊢ <Q ⊆ (Q × Q)
2	1	brel 4827	. . . 4 ⊢ (𝑠 <Q 𝑡 → (𝑠 ∈ Q ∧ 𝑡 ∈ Q))
3	2	simpld 112	. . 3 ⊢ (𝑠 <Q 𝑡 → 𝑠 ∈ Q)
4	3	3ad2ant2 1050	. 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 8052	. . . . . 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 1051	. . . 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 1068	. . . . . . . . 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 7767	. . . . . . . . . . 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 492	. . . . . . . . . 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 1050	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑡 ∈ Q)
15	14	ad2antrr 492	. . . . . . . . . 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 533	. . . . . . . . . . 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 7789	. . . . . . . . . . 11 ⊢ (𝑏 ∈ N → [⟨𝑏, 1o⟩] ~Q ∈ Q)
18		recclnq 7759	. . . . . . . . . . 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 7748	. . . . . . . . . . 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 6259	. . . . . . . . 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 7961	. . . . . . . 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 7963	. . . . . . . 8 ⊢ <P Or P
27		ltrelpr 7872	. . . . . . . 8 ⊢ <P ⊆ (P × P)
28	26, 27	sotri 5183	. . . . . . 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 424	. . . . . 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 2652	. . . 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 3904	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑟 → ⟨𝑏, 1o⟩ = ⟨𝑟, 1o⟩)
34	33	eceq1d 6843	. . . . . . . . . 10 ⊢ (𝑏 = 𝑟 → [⟨𝑏, 1o⟩] ~Q = [⟨𝑟, 1o⟩] ~Q )
35	34	fveq2d 5699	. . . . . . . . 9 ⊢ (𝑏 = 𝑟 → (*Q‘[⟨𝑏, 1o⟩] ~Q ) = (*Q‘[⟨𝑟, 1o⟩] ~Q ))
36	35	oveq2d 6101	. . . . . . . 8 ⊢ (𝑏 = 𝑟 → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
37	36	breq2d 4142	. . . . . . 7 ⊢ (𝑏 = 𝑟 → (𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))))
38	37	abbidv 2358	. . . . . 6 ⊢ (𝑏 = 𝑟 → {𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))})
39	36	breq1d 4140	. . . . . . 7 ⊢ (𝑏 = 𝑟 → ((𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞))
40	39	abbidv 2358	. . . . . 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 3912	. . . . 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 5695	. . . . 5 ⊢ (𝑏 = 𝑟 → (𝐹‘𝑏) = (𝐹‘𝑟))
43	41, 42	breq12d 4143	. . . 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 2787	. . 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 8052	. 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 421	1 ⊢ ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  {crab 2532  ⟨cop 3712   class class class wbr 4130  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  1^st c1st 6372  1_oc1o 6680  [cec 6805  Ncnpi 7639   <_N clti 7642   ~_Q ceq 7646  Qcnq 7647   +_Q cplq 7649  *_Qcrq 7651   <_Q cltq 7652  Pcnp 7658   +_P cpp 7660  <_P cltp 7662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-inp 7833  df-iltp 7837

This theorem is used by:  caucvgprprlemrnd  8068