Theorem caucvgprprlemopl 6887

Description: Lemma for caucvgprpr 6902. 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 6875	. . . 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 269	. . 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 271	. 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 471	. . . . . 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 497	. . . . . 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 5324	. . . . 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 6665	. . . . 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 498	. . . . 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 6875	. . . . . . . . 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 268	. . . . . . . 8 ⊢ (𝑠 ∈ (1st ‘𝐿) → 𝑠 ∈ Q)
14	13	ad2antlr 472	. . . . . . 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 6612	. . . . . . . . 9 ⊢ (𝑏 ∈ N → [⟨𝑏, 1𝑜⟩] ~Q ∈ Q)
16		recclnq 6582	. . . . . . . . 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 473	. . . . . . 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 6565	. . . . . . 7 ⊢ ((𝑠 ∈ Q ∧ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q)
20	14, 18, 19	syl2anc 403	. . . . . 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 6741	. . . . . 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 403	. . . . 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 165	. . . 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 6678	. . . 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 403	. . 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 270	. . . . . . . 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 270	. . . . . . . 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 6597	. . . . . . . 8 ⊢ (((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q ∧ 𝑠 ∈ Q) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠))
29	26, 27, 28	syl2anc 403	. . . . . . 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 6571	. . . . . . . 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 403	. . . . . . 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 3809	. . . . . 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 498	. . . . . 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 6588	. . . . . . 7 ⊢ <Q Or Q
35		ltrelnq 6555	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
36	34, 35	sotri 4740	. . . . . 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 403	. . . . 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 270	. . . . . . 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 497	. . . . . . 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 6671	. . . . . . 7 ⊢ ((⟨(1st ‘(𝐹‘𝑏)), (2nd ‘(𝐹‘𝑏))⟩ ∈ P ∧ 𝑎 ∈ (1st ‘(𝐹‘𝑏))) → 𝑎 ∈ Q)
41	38, 39, 40	syl2anc 403	. . . . . 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 6598	. . . . . 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 403	. . . . 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 145	. . . 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 471	. . . . . . . . . . 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 471	. . . . . . . . . . 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 6571	. . . . . . . . . . 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 403	. . . . . . . . . 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 471	. . . . . . . . . 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 3807	. . . . . . . . 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 108	. . . . . . . . 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 3811	. . . . . . . 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 496	. . . . . . . . 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 6590	. . . . . . . . 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 1169	. . . . . . . 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 165	. . . . . . 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 475	. . . . . . . . 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 6571	. . . . . . . . . . . . 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 403	. . . . . . . . . . . 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 2115	. . . . . . . . . . 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 471	. . . . . . . . . . 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 2155	. . . . . . . . . 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 6565	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ Q ∧ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q) → (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q)
64	53, 46, 63	syl2anc 403	. . . . . . . . . . 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 475	. . . . . . . . . . 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 6741	. . . . . . . . . . 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 403	. . . . . . . . . 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 145	. . . . . . . . 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 3570	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑏 → ⟨𝑟, 1𝑜⟩ = ⟨𝑏, 1𝑜⟩)
70	69	eceq1d 6165	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑏 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝑏, 1𝑜⟩] ~Q )
71	70	fveq2d 5202	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))
72	71	oveq2d 5548	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑏 → (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) = (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
73	72	breq2d 3797	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → (𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))))
74	73	abbidv 2196	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → {𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))})
75	72	breq1d 3795	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → ((𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞))
76	75	abbidv 2196	. . . . . . . . . . . 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 3578	. . . . . . . . . . 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 5198	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑏 → (𝐹‘𝑟) = (𝐹‘𝑏))
79	77, 78	breq12d 3798	. . . . . . . . . 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 2701	. . . . . . . . 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 403	. . . . . . . 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 6875	. . . . . . . 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 408	. . . . . . 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 300	. . . . . 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 113	. . . . 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 2463	. . . 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 2481	. 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 2481	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  {cab 2067  ∀wral 2348  ∃wrex 2349  {crab 2352  ⟨cop 3401   class class class wbr 3785  ⟶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  Pcnp 6481   +_P cpp 6483  <_P cltp 6485

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  df-inp 6656  df-iltp 6660

This theorem is referenced by:  caucvgprprlemrnd  6891