Theorem caucvgprprlemml 8014

Description: Lemma for caucvgprpr 8032. 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 5672	. . . . . 6 ⊢ (𝑚 = 1o → (𝐹‘𝑚) = (𝐹‘1o))
2	1	breq2d 4123	. . . . 5 ⊢ (𝑚 = 1o → (𝐴<P (𝐹‘𝑚) ↔ 𝐴<P (𝐹‘1o)))
3		caucvgprpr.bnd	. . . . 5 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))
4		1pi 7635	. . . . . 6 ⊢ 1o ∈ N
5	4	a1i 9	. . . . 5 ⊢ (𝜑 → 1o ∈ N)
6	2, 3, 5	rspcdva 2928	. . . 4 ⊢ (𝜑 → 𝐴<P (𝐹‘1o))
7		ltrelpr 7825	. . . . . 6 ⊢ <P ⊆ (P × P)
8	7	brel 4804	. . . . 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 7795	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
12		prml 7797	. . . 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 7734	. . . 4 ⊢ (𝑥 ∈ Q → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) <Q 𝑥)
16	15	ad2antrl 490	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) <Q 𝑥)
17		simplr 529	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) <Q 𝑥) → 𝑠 ∈ Q)
18		archrecnq 7983	. . . . . . . 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 529	. . . . . . . . . . . . . . . 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 7742	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ N → [⟨𝑟, 1o⟩] ~Q ∈ Q)
23		recclnq 7712	. . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 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 7720	. . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . 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 536	. . . . . . . . . . . . 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 7718	. . . . . . . . . . . . . 14 ⊢ <Q Or Q
31		ltrelnq 7685	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
32	30, 31	sotri 5160	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 496	. . . . . . . . . . . . 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 533	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → 𝑥 ∈ (1st ‘𝐴))
36	35	ad4antr 494	. . . . . . . . . . . . 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 7804	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 7695	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ Q ∧ (*Q‘[⟨𝑟, 1o⟩] ~Q ) ∈ Q) → (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ∈ Q)
42	25, 24, 41	syl2anc 411	. . . . . . . . . . . 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 7871	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 5672	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑟 → (𝐹‘𝑚) = (𝐹‘𝑟))
47	46	breq2d 4123	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑟 → (𝐴<P (𝐹‘𝑚) ↔ 𝐴<P (𝐹‘𝑟)))
48	3	ad5antr 496	. . . . . . . . . . 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 2928	. . . . . . . . . 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 7916	. . . . . . . . . . 11 ⊢ <P Or P
51	50, 7	sotri 5160	. . . . . . . . . 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 411	. . . . . . . . 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 2646	. . . . . . 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 6059	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
57	56	breq2d 4123	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑠 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))))
58	57	abbidv 2354	. . . . . . . . . 10 ⊢ (𝑙 = 𝑠 → {𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))})
59	56	breq1d 4121	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑠 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞))
60	59	abbidv 2354	. . . . . . . . . 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 3893	. . . . . . . . 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 4121	. . . . . . . 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 2545	. . . . . . 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 5675	. . . . . . . 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 7683	. . . . . . . . . 10 ⊢ Q ∈ V
67	66	rabex 4258	. . . . . . . . 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 4258	. . . . . . . . 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 6342	. . . . . . . 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 2255	. . . . . . 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 2978	. . . . . 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 417	. . . . 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 2646	. . 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 2667	1 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃wrex 2523  {crab 2526  ⟨cop 3694   class class class wbr 4111  ⟶wf 5350  ‘cfv 5354  (class class class)co 6052  1^st c1st 6334  2^nd c2nd 6335  1_oc1o 6642  [cec 6767  Ncnpi 7592   <_N clti 7595   ~_Q ceq 7599  Qcnq 7600   +_Q cplq 7602  *_Qcrq 7604   <_Q cltq 7605  Pcnp 7611   +_P cpp 7613  <_P cltp 7615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7624  df-pli 7625  df-mi 7626  df-lti 7627  df-plpq 7664  df-mpq 7665  df-enq 7667  df-nqqs 7668  df-plqqs 7669  df-mqqs 7670  df-1nqqs 7671  df-rq 7672  df-ltnqqs 7673  df-inp 7786  df-iltp 7790

This theorem is referenced by:  caucvgprprlemm  8016