Theorem caucvgprprlemopl 6795

Description: Lemma for caucvgprpr 6810. 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 6783	. . . 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 260	. . 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 262	. 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 457	. . . . . 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 483	. . . . . 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 5303	. . . . 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 6573	. . . . 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 484	. . . . 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 6783	. . . . . . . . 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 259	. . . . . . . 8 ⊢ (𝑠 ∈ (1st ‘𝐿) → 𝑠 ∈ Q)
14	13	ad2antlr 458	. . . . . . 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 6520	. . . . . . . . 9 ⊢ (𝑏 ∈ N → [⟨𝑏, 1𝑜⟩] ~Q ∈ Q)
16		recclnq 6490	. . . . . . . . 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 459	. . . . . . 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 6473	. . . . . . 7 ⊢ ((𝑠 ∈ Q ∧ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q)
20	14, 18, 19	syl2anc 391	. . . . . 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 6649	. . . . . 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 391	. . . . 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 156	. . . 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 6586	. . . 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 391	. . 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 261	. . . . . . . 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 261	. . . . . . . 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 6505	. . . . . . . 8 ⊢ (((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q ∧ 𝑠 ∈ Q) → (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q ((*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) +Q 𝑠))
29	26, 27, 28	syl2anc 391	. . . . . . 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 6479	. . . . . . . 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 391	. . . . . . 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 3788	. . . . . 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 484	. . . . . 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 6496	. . . . . . 7 ⊢ <Q Or Q
35		ltrelnq 6463	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
36	34, 35	sotri 4720	. . . . . 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 391	. . . . 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 261	. . . . . . 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 483	. . . . . . 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 6579	. . . . . . 7 ⊢ ((⟨(1st ‘(𝐹‘𝑏)), (2nd ‘(𝐹‘𝑏))⟩ ∈ P ∧ 𝑎 ∈ (1st ‘(𝐹‘𝑏))) → 𝑎 ∈ Q)
41	38, 39, 40	syl2anc 391	. . . . . 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 6506	. . . . . 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 391	. . . . 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 135	. . . 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 457	. . . . . . . . . . 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 457	. . . . . . . . . . 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 6479	. . . . . . . . . . 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 391	. . . . . . . . . 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 457	. . . . . . . . . 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 3786	. . . . . . . . 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 103	. . . . . . . . 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 3790	. . . . . . . 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 482	. . . . . . . . 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 6498	. . . . . . . . 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 1135	. . . . . . . 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 156	. . . . . . 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 461	. . . . . . . . 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 6479	. . . . . . . . . . . . 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 391	. . . . . . . . . . . 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 2074	. . . . . . . . . . 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 457	. . . . . . . . . . 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 2114	. . . . . . . . . 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 6473	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ Q ∧ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) ∈ Q) → (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) ∈ Q)
64	53, 46, 63	syl2anc 391	. . . . . . . . . . 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 461	. . . . . . . . . . 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 6649	. . . . . . . . . . 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 391	. . . . . . . . . 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 135	. . . . . . . . 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 3549	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = 𝑏 → ⟨𝑟, 1𝑜⟩ = ⟨𝑏, 1𝑜⟩)
70	69	eceq1d 6142	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑏 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝑏, 1𝑜⟩] ~Q )
71	70	fveq2d 5182	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑏 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))
72	71	oveq2d 5528	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑏 → (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) = (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )))
73	72	breq2d 3776	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → (𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))))
74	73	abbidv 2155	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑏 → {𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))})
75	72	breq1d 3774	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑏 → ((𝑡 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑡 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞))
76	75	abbidv 2155	. . . . . . . . . . . 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 3557	. . . . . . . . . . 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 5178	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑏 → (𝐹‘𝑟) = (𝐹‘𝑏))
79	77, 78	breq12d 3777	. . . . . . . . . 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 2656	. . . . . . . . 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 391	. . . . . . . 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 6783	. . . . . . . 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 394	. . . . . . 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 290	. . . . . 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 108	. . . . 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 2421	. . . 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 2437	. 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 2437	1 ⊢ ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  {crab 2310  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1^st c1st 5765  2^nd c2nd 5766  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   <_N clti 6373   ~_Q ceq 6377  Qcnq 6378   +_Q cplq 6380  *_Qcrq 6382   <_Q cltq 6383  Pcnp 6389   +_P cpp 6391  <_P cltp 6393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564  df-iltp 6568

This theorem is referenced by:  caucvgprprlemrnd  6799