Theorem caucvgprprlemml 8068

Description: Lemma for caucvgprpr 8086. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-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
caucvgprprlemml	⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿))

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

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

Proof of Theorem caucvgprprlemml

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5698	. . . . . 6 ⊢ (𝑚 = 1o → (𝐹‘𝑚) = (𝐹‘1o))
2	1	breq2d 4143	. . . . 5 ⊢ (𝑚 = 1o → (𝐴<P (𝐹‘𝑚) ↔ 𝐴<P (𝐹‘1o)))
3		caucvgprpr.bnd	. . . . 5 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))
4		1pi 7689	. . . . . 6 ⊢ 1o ∈ N
5	4	a1i 9	. . . . 5 ⊢ (𝜑 → 1o ∈ N)
6	2, 3, 5	rspcdva 2934	. . . 4 ⊢ (𝜑 → 𝐴<P (𝐹‘1o))
7		ltrelpr 7879	. . . . . 6 ⊢ <P ⊆ (P × P)
8	7	brel 4829	. . . . 5 ⊢ (𝐴<P (𝐹‘1o) → (𝐴 ∈ P ∧ (𝐹‘1o) ∈ P))
9	8	simpld 112	. . . 4 ⊢ (𝐴<P (𝐹‘1o) → 𝐴 ∈ P)
10	6, 9	syl 14	. . 3 ⊢ (𝜑 → 𝐴 ∈ P)
11		prop 7849	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
12		prml 7851	. . . 4 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘𝐴))
13	11, 12	syl 14	. . 3 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘𝐴))
14	10, 13	syl 14	. 2 ⊢ (𝜑 → ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘𝐴))
15		subhalfnqq 7788	. . . 4 ⊢ (𝑥 ∈ Q → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) <Q 𝑥)
16	15	ad2antrl 494	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) <Q 𝑥)
17		simplr 533	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → 𝑠 ∈ Q)
18		archrecnq 8037	. . . . . . . 8 ⊢ (𝑠 ∈ Q → ∃𝑟 ∈ N (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠)
19	17, 18	syl 14	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → ∃𝑟 ∈ N (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠)
20		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠)
21		simplr 533	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝑟 ∈ N)
22		nnnq 7796	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ N → [⟨𝑟, 1o⟩] ~Q ∈ Q)
23		recclnq 7766	. . . . . . . . . . . . . . . 16 ⊢ ([⟨𝑟, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝑟, 1o⟩] ~Q ) ∈ Q)
24	21, 22, 23	3syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (*Q‘[⟨𝑟, 1o⟩] ~Q ) ∈ Q)
25	17	ad2antrr 492	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝑠 ∈ Q)
26		ltanqg 7774	. . . . . . . . . . . . . . 15 ⊢ (((*Q‘[⟨𝑟, 1o⟩] ~Q ) ∈ Q ∧ 𝑠 ∈ Q ∧ 𝑠 ∈ Q) → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q (𝑠 +Q 𝑠)))
27	24, 25, 25, 26	syl3anc 1278	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q (𝑠 +Q 𝑠)))
28	20, 27	mpbid 147	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q (𝑠 +Q 𝑠))
29		simpllr 540	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q 𝑠) <Q 𝑥)
30		ltsonq 7772	. . . . . . . . . . . . . 14 ⊢ <Q Or Q
31		ltrelnq 7739	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
32	30, 31	sotri 5186	. . . . . . . . . . . . 13 ⊢ (((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q (𝑠 +Q 𝑠) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥)
33	28, 29, 32	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥)
34	10	ad5antr 500	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝐴 ∈ P)
35		simprr 537	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝑥 ∈ (1st ‘𝐴))
36	35	ad4antr 498	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝑥 ∈ (1st ‘𝐴))
37		prcdnql 7858	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st ‘𝐴)))
38	11, 37	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st ‘𝐴)))
39	34, 36, 38	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑥 → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st ‘𝐴)))
40	33, 39	mpd 13	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st ‘𝐴))
41		addclnq 7749	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ Q ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ Q)
42	25, 24, 41	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ Q)
43		nqprl 7925	. . . . . . . . . . . 12 ⊢ (((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ Q ∧ 𝐴 ∈ P) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st ‘𝐴) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P 𝐴))
44	42, 34, 43	syl2anc 415	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ((𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ (1st ‘𝐴) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P 𝐴))
45	40, 44	mpbid 147	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P 𝐴)
46		fveq2 5698	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑟 → (𝐹‘𝑚) = (𝐹‘𝑟))
47	46	breq2d 4143	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑟 → (𝐴<P (𝐹‘𝑚) ↔ 𝐴<P (𝐹‘𝑟)))
48	3	ad5antr 500	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))
49	47, 48, 21	rspcdva 2934	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → 𝐴<P (𝐹‘𝑟))
50		ltsopr 7970	. . . . . . . . . . 11 ⊢ <P Or P
51	50, 7	sotri 5186	. . . . . . . . . 10 ⊢ ((⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P 𝐴 ∧ 𝐴<P (𝐹‘𝑟)) → ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟))
52	45, 49, 51	syl2anc 415	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠) → ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟))
53	52	ex 115	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) ∧ 𝑟 ∈ N) → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠 → ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
54	53	reximdva 2652	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → (∃𝑟 ∈ N (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑠 → ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
55	19, 54	mpd 13	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟))
56		oveq1 6095	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
57	56	breq2d 4143	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑠 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))))
58	57	abbidv 2358	. . . . . . . . . 10 ⊢ (𝑙 = 𝑠 → {𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))})
59	56	breq1d 4141	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞))
60	59	abbidv 2358	. . . . . . . . . 10 ⊢ (𝑙 = 𝑠 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞})
61	58, 60	opeq12d 3912	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩)
62	61	breq1d 4141	. . . . . . . 8 ⊢ (𝑙 = 𝑠 → (⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
63	62	rexbidv 2551	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
64		caucvgprpr.lim	. . . . . . . . 9 ⊢ 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}⟩}⟩
65	64	fveq2i 5701	. . . . . . . 8 ⊢ (1st ‘𝐿) = (1st ‘⟨{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}⟩}⟩)
66		nqex 7737	. . . . . . . . . 10 ⊢ Q ∈ V
67	66	rabex 4282	. . . . . . . . 9 ⊢ {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)} ∈ V
68	66	rabex 4282	. . . . . . . . 9 ⊢ {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}⟩} ∈ V
69	67, 68	op1st 6384	. . . . . . . 8 ⊢ (1st ‘⟨{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}⟩}⟩) = {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)}
70	65, 69	eqtri 2259	. . . . . . 7 ⊢ (1st ‘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)}
71	63, 70	elrab2 2985	. . . . . 6 ⊢ (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)))
72	17, 55, 71	sylanbrc 421	. . . . 5 ⊢ ((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → 𝑠 ∈ (1st ‘𝐿))
73	72	ex 115	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) → ((𝑠 +Q 𝑠) <Q 𝑥 → 𝑠 ∈ (1st ‘𝐿)))
74	73	reximdva 2652	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → (∃𝑠 ∈ Q (𝑠 +Q 𝑠) <Q 𝑥 → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿)))
75	16, 74	mpd 13	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿))
76	14, 75	rexlimddv 2673	1 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  {crab 2532  ⟨cop 3712   class class class wbr 4131  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088  1^st c1st 6376  2^nd c2nd 6377  1_oc1o 6684  [cec 6809  Ncnpi 7646   <_N clti 7649   ~_Q ceq 7653  Qcnq 7654   +_Q cplq 7656  *_Qcrq 7658   <_Q cltq 7659  Pcnp 7665   +_P cpp 7667  <_P cltp 7669

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7678  df-pli 7679  df-mi 7680  df-lti 7681  df-plpq 7718  df-mpq 7719  df-enq 7721  df-nqqs 7722  df-plqqs 7723  df-mqqs 7724  df-1nqqs 7725  df-rq 7726  df-ltnqqs 7727  df-inp 7840  df-iltp 7844

This theorem is used by:  caucvgprprlemm  8070