Theorem caucvgprprlemopl 7659

Description: Lemma for caucvgprpr 7674. The lower cut of the putative limit is open. (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
caucvgprprlemopl	⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))

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

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

Proof of Theorem caucvgprprlemopl

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprpr.lim	. . . . 5 ⊢ 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}⟩}⟩
2	1	caucvgprprlemell 7647	. . . 4 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
3	2	simprbi 273	. . 3 ⊢ (𝑠 ∈ (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))
4	3	adantl 275	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (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))
5		caucvgprpr.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:N⟶P)
6	5	ad2antrr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → 𝐹:N⟶P)
7		simprl 526	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → 𝑏 ∈ N)
8	6, 7	ffvelrnd 5632	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → (𝐹‘𝑏) ∈ P)
9		prop 7437	. . . . 5 ⊢ ((𝐹‘𝑏) ∈ P → ⟨(1st ‘(𝐹‘𝑏)), (2nd ‘(𝐹‘𝑏))⟩ ∈ P)
10	8, 9	syl 14	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → ⟨(1st ‘(𝐹‘𝑏)), (2nd ‘(𝐹‘𝑏))⟩ ∈ P)
11		simprr 527	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏))
12	1	caucvgprprlemell 7647	. . . . . . . . 9 ⊢ (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)))
13	12	simplbi 272	. . . . . . . 8 ⊢ (𝑠 ∈ (1st ‘𝐿) → 𝑠 ∈ Q)
14	13	ad2antlr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → 𝑠 ∈ Q)
15		nnnq 7384	. . . . . . . . 9 ⊢ (𝑏 ∈ N → [⟨𝑏, 1o⟩] ~Q ∈ Q)
16		recclnq 7354	. . . . . . . . 9 ⊢ ([⟨𝑏, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
17	15, 16	syl 14	. . . . . . . 8 ⊢ (𝑏 ∈ N → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
18	17	ad2antrl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
19		addclnq 7337	. . . . . . 7 ⊢ ((𝑠 ∈ Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
20	14, 18, 19	syl2anc 409	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
21		nqprl 7513	. . . . . 6 ⊢ (((𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q ∧ (𝐹‘𝑏) ∈ P) → ((𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ (1st ‘(𝐹‘𝑏)) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
22	20, 8, 21	syl2anc 409	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → ((𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ (1st ‘(𝐹‘𝑏)) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
23	11, 22	mpbird 166	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ (1st ‘(𝐹‘𝑏)))
24		prnmaxl 7450	. . . 4 ⊢ ((⟨(1st ‘(𝐹‘𝑏)), (2nd ‘(𝐹‘𝑏))⟩ ∈ P ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ (1st ‘(𝐹‘𝑏))) → ∃𝑎 ∈ (1st ‘(𝐹‘𝑏))(𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)
25	10, 23, 24	syl2anc 409	. . 3 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → ∃𝑎 ∈ (1st ‘(𝐹‘𝑏))(𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)
26	18	adantr 274	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
27	14	adantr 274	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → 𝑠 ∈ Q)
28		ltaddnq 7369	. . . . . . . 8 ⊢ (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q ∧ 𝑠 ∈ Q) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠))
29	26, 27, 28	syl2anc 409	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠))
30		addcomnqg 7343	. . . . . . . 8 ⊢ (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q ∧ 𝑠 ∈ Q) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
31	26, 27, 30	syl2anc 409	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠) = (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
32	29, 31	breqtrd 4015	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
33		simprr 527	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)
34		ltsonq 7360	. . . . . . 7 ⊢ <Q Or Q
35		ltrelnq 7327	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
36	34, 35	sotri 5006	. . . . . 6 ⊢ (((*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑎)
37	32, 33, 36	syl2anc 409	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑎)
38	10	adantr 274	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → ⟨(1st ‘(𝐹‘𝑏)), (2nd ‘(𝐹‘𝑏))⟩ ∈ P)
39		simprl 526	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → 𝑎 ∈ (1st ‘(𝐹‘𝑏)))
40		elprnql 7443	. . . . . . 7 ⊢ ((⟨(1st ‘(𝐹‘𝑏)), (2nd ‘(𝐹‘𝑏))⟩ ∈ P ∧ 𝑎 ∈ (1st ‘(𝐹‘𝑏))) → 𝑎 ∈ Q)
41	38, 39, 40	syl2anc 409	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → 𝑎 ∈ Q)
42		ltexnqq 7370	. . . . . 6 ⊢ (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q ∧ 𝑎 ∈ Q) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑎 ↔ ∃𝑡 ∈ Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎))
43	26, 41, 42	syl2anc 409	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑎 ↔ ∃𝑡 ∈ Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎))
44	37, 43	mpbid 146	. . . 4 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → ∃𝑡 ∈ Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎)
45	27	ad2antrr 485	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑠 ∈ Q)
46	26	ad2antrr 485	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q)
47		addcomnqg 7343	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) = ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠))
48	45, 46, 47	syl2anc 409	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) = ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠))
49	33	ad2antrr 485	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)
50	48, 49	eqbrtrrd 4013	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠) <Q 𝑎)
51		simpr 109	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎)
52	50, 51	breqtrrd 4017	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡))
53		simplr 525	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑡 ∈ Q)
54		ltanqg 7362	. . . . . . . . 9 ⊢ ((𝑠 ∈ Q ∧ 𝑡 ∈ Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → (𝑠 <Q 𝑡 ↔ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡)))
55	45, 53, 46, 54	syl3anc 1233	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑠 <Q 𝑡 ↔ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑠) <Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡)))
56	52, 55	mpbird 166	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑠 <Q 𝑡)
57	7	ad3antrrr 489	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑏 ∈ N)
58		addcomnqg 7343	. . . . . . . . . . . . 13 ⊢ (((*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q ∧ 𝑡 ∈ Q) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
59	46, 53, 58	syl2anc 409	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
60	59, 51	eqtr3d 2205	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) = 𝑎)
61	39	ad2antrr 485	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑎 ∈ (1st ‘(𝐹‘𝑏)))
62	60, 61	eqeltrd 2247	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ (1st ‘(𝐹‘𝑏)))
63		addclnq 7337	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ Q ∧ (*Q‘[⟨𝑏, 1o⟩] ~Q ) ∈ Q) → (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
64	53, 46, 63	syl2anc 409	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q)
65	8	ad3antrrr 489	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝐹‘𝑏) ∈ P)
66		nqprl 7513	. . . . . . . . . . 11 ⊢ (((𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ Q ∧ (𝐹‘𝑏) ∈ P) → ((𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ (1st ‘(𝐹‘𝑏)) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
67	64, 65, 66	syl2anc 409	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → ((𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) ∈ (1st ‘(𝐹‘𝑏)) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
68	62, 67	mpbid 146	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏))
69		opeq1 3765	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑏 → ⟨𝑟, 1o⟩ = ⟨𝑏, 1o⟩)
70	69	eceq1d 6549	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑏 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
71	70	fveq2d 5500	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
72	71	oveq2d 5869	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑏 → (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
73	72	breq2d 4001	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → (𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))))
74	73	abbidv 2288	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → {𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))})
75	72	breq1d 3999	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → ((𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞))
76	75	abbidv 2288	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞})
77	74, 76	opeq12d 3773	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑏 → ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩)
78		fveq2 5496	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑏 → (𝐹‘𝑟) = (𝐹‘𝑏))
79	77, 78	breq12d 4002	. . . . . . . . . 10 ⊢ (𝑟 = 𝑏 → (⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
80	79	rspcev 2834	. . . . . . . . 9 ⊢ ((𝑏 ∈ N ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)) → ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟))
81	57, 68, 80	syl2anc 409	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟))
82	1	caucvgprprlemell 7647	. . . . . . . 8 ⊢ (𝑡 ∈ (1st ‘𝐿) ↔ (𝑡 ∈ Q ∧ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑡 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
83	53, 81, 82	sylanbrc 415	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → 𝑡 ∈ (1st ‘𝐿))
84	56, 83	jca 304	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) ∧ ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎) → (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))
85	84	ex 114	. . . . 5 ⊢ (((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) ∧ 𝑡 ∈ Q) → (((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎 → (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿))))
86	85	reximdva 2572	. . . 4 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → (∃𝑡 ∈ Q ((*Q‘[⟨𝑏, 1o⟩] ~Q ) +Q 𝑡) = 𝑎 → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿))))
87	44, 86	mpd 13	. . 3 ⊢ ((((𝜑 ∧ 𝑠 ∈ (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))) ∧ (𝑎 ∈ (1st ‘(𝐹‘𝑏)) ∧ (𝑠 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑎)) → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))
88	25, 87	rexlimddv 2592	. 2 ⊢ (((𝜑 ∧ 𝑠 ∈ (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))) → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))
89	4, 88	rexlimddv 2592	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  {crab 2452  ⟨cop 3586   class class class wbr 3989  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  1^st c1st 6117  2^nd c2nd 6118  1_oc1o 6388  [cec 6511  Ncnpi 7234   <_N clti 7237   ~_Q ceq 7241  Qcnq 7242   +_Q cplq 7244  *_Qcrq 7246   <_Q cltq 7247  Pcnp 7253   +_P cpp 7255  <_P cltp 7257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-iltp 7432

This theorem is referenced by:  caucvgprprlemrnd  7663