Theorem caucvgprprlemmu 7779

Description: Lemma for caucvgprpr 7796. The upper 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
caucvgprprlemmu	⊢ (𝜑 → ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿))

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

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

Proof of Theorem caucvgprprlemmu

Dummy variables 𝑓 𝑔 ℎ 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprpr.f	. . . 4 ⊢ (𝜑 → 𝐹:N⟶P)
2		1pi 7399	. . . . 5 ⊢ 1o ∈ N
3	2	a1i 9	. . . 4 ⊢ (𝜑 → 1o ∈ N)
4	1, 3	ffvelcdmd 5701	. . 3 ⊢ (𝜑 → (𝐹‘1o) ∈ P)
5		prop 7559	. . 3 ⊢ ((𝐹‘1o) ∈ P → ⟨(1st ‘(𝐹‘1o)), (2nd ‘(𝐹‘1o))⟩ ∈ P)
6		prmu 7562	. . 3 ⊢ (⟨(1st ‘(𝐹‘1o)), (2nd ‘(𝐹‘1o))⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (2nd ‘(𝐹‘1o)))
7	4, 5, 6	3syl 17	. 2 ⊢ (𝜑 → ∃𝑥 ∈ Q 𝑥 ∈ (2nd ‘(𝐹‘1o)))
8		simprl 529	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → 𝑥 ∈ Q)
9		1nq 7450	. . . 4 ⊢ 1Q ∈ Q
10		addclnq 7459	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 1Q ∈ Q) → (𝑥 +Q 1Q) ∈ Q)
11	8, 9, 10	sylancl 413	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝑥 +Q 1Q) ∈ Q)
12	2	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → 1o ∈ N)
13		simprr 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → 𝑥 ∈ (2nd ‘(𝐹‘1o)))
14	4	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝐹‘1o) ∈ P)
15		nqpru 7636	. . . . . . . . 9 ⊢ ((𝑥 ∈ Q ∧ (𝐹‘1o) ∈ P) → (𝑥 ∈ (2nd ‘(𝐹‘1o)) ↔ (𝐹‘1o)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
16	8, 14, 15	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝑥 ∈ (2nd ‘(𝐹‘1o)) ↔ (𝐹‘1o)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩))
17	13, 16	mpbid 147	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝐹‘1o)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩)
18		ltaprg 7703	. . . . . . . . 9 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
19	18	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
20		nqprlu 7631	. . . . . . . . 9 ⊢ (𝑥 ∈ Q → ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩ ∈ P)
21	8, 20	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩ ∈ P)
22		nqprlu 7631	. . . . . . . . 9 ⊢ (1Q ∈ Q → ⟨{𝑝 ∣ 𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩ ∈ P)
23	9, 22	mp1i 10	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → ⟨{𝑝 ∣ 𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩ ∈ P)
24		addcomprg 7662	. . . . . . . . 9 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
25	24	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
26	19, 14, 21, 23, 25	caovord2d 6097	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → ((𝐹‘1o)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩ ↔ ((𝐹‘1o) +P ⟨{𝑝 ∣ 𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩)<P (⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩)))
27	17, 26	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → ((𝐹‘1o) +P ⟨{𝑝 ∣ 𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩)<P (⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩))
28		df-1nqqs 7435	. . . . . . . . . . . . 13 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
29	28	fveq2i 5564	. . . . . . . . . . . 12 ⊢ (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
30		rec1nq 7479	. . . . . . . . . . . 12 ⊢ (*Q‘1Q) = 1Q
31	29, 30	eqtr3i 2219	. . . . . . . . . . 11 ⊢ (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
32	31	breq2i 4042	. . . . . . . . . 10 ⊢ (𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q ) ↔ 𝑝 <Q 1Q)
33	32	abbii 2312	. . . . . . . . 9 ⊢ {𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q 1Q}
34	31	breq1i 4041	. . . . . . . . . 10 ⊢ ((*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞 ↔ 1Q <Q 𝑞)
35	34	abbii 2312	. . . . . . . . 9 ⊢ {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ 1Q <Q 𝑞}
36	33, 35	opeq12i 3814	. . . . . . . 8 ⊢ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩
37	36	oveq2i 5936	. . . . . . 7 ⊢ ((𝐹‘1o) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘1o) +P ⟨{𝑝 ∣ 𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩)
38	37	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → ((𝐹‘1o) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘1o) +P ⟨{𝑝 ∣ 𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩))
39		addnqpr 7645	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 1Q ∈ Q) → ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩ = (⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩))
40	8, 9, 39	sylancl 413	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩ = (⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q 1Q}, {𝑞 ∣ 1Q <Q 𝑞}⟩))
41	27, 38, 40	3brtr4d 4066	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → ((𝐹‘1o) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩)
42		fveq2 5561	. . . . . . . 8 ⊢ (𝑟 = 1o → (𝐹‘𝑟) = (𝐹‘1o))
43		opeq1 3809	. . . . . . . . . . . . 13 ⊢ (𝑟 = 1o → ⟨𝑟, 1o⟩ = ⟨1o, 1o⟩)
44	43	eceq1d 6637	. . . . . . . . . . . 12 ⊢ (𝑟 = 1o → [⟨𝑟, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
45	44	fveq2d 5565	. . . . . . . . . . 11 ⊢ (𝑟 = 1o → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
46	45	breq2d 4046	. . . . . . . . . 10 ⊢ (𝑟 = 1o → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )))
47	46	abbidv 2314	. . . . . . . . 9 ⊢ (𝑟 = 1o → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )})
48	45	breq1d 4044	. . . . . . . . . 10 ⊢ (𝑟 = 1o → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞))
49	48	abbidv 2314	. . . . . . . . 9 ⊢ (𝑟 = 1o → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞})
50	47, 49	opeq12d 3817	. . . . . . . 8 ⊢ (𝑟 = 1o → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞}⟩)
51	42, 50	oveq12d 5943	. . . . . . 7 ⊢ (𝑟 = 1o → ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘1o) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞}⟩))
52	51	breq1d 4044	. . . . . 6 ⊢ (𝑟 = 1o → (((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩ ↔ ((𝐹‘1o) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩))
53	52	rspcev 2868	. . . . 5 ⊢ ((1o ∈ N ∧ ((𝐹‘1o) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩) → ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩)
54	12, 41, 53	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩)
55		breq2 4038	. . . . . . . . 9 ⊢ (𝑢 = (𝑥 +Q 1Q) → (𝑝 <Q 𝑢 ↔ 𝑝 <Q (𝑥 +Q 1Q)))
56	55	abbidv 2314	. . . . . . . 8 ⊢ (𝑢 = (𝑥 +Q 1Q) → {𝑝 ∣ 𝑝 <Q 𝑢} = {𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)})
57		breq1 4037	. . . . . . . . 9 ⊢ (𝑢 = (𝑥 +Q 1Q) → (𝑢 <Q 𝑞 ↔ (𝑥 +Q 1Q) <Q 𝑞))
58	57	abbidv 2314	. . . . . . . 8 ⊢ (𝑢 = (𝑥 +Q 1Q) → {𝑞 ∣ 𝑢 <Q 𝑞} = {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞})
59	56, 58	opeq12d 3817	. . . . . . 7 ⊢ (𝑢 = (𝑥 +Q 1Q) → ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩)
60	59	breq2d 4046	. . . . . 6 ⊢ (𝑢 = (𝑥 +Q 1Q) → (((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}⟩ ↔ ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩))
61	60	rexbidv 2498	. . . . 5 ⊢ (𝑢 = (𝑥 +Q 1Q) → (∃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}⟩ ↔ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩))
62		caucvgprpr.lim	. . . . . . 7 ⊢ 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}⟩}⟩
63	62	fveq2i 5564	. . . . . 6 ⊢ (2nd ‘𝐿) = (2nd ‘⟨{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}⟩}⟩)
64		nqex 7447	. . . . . . . 8 ⊢ Q ∈ V
65	64	rabex 4178	. . . . . . 7 ⊢ {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
66	64	rabex 4178	. . . . . . 7 ⊢ {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
67	65, 66	op2nd 6214	. . . . . 6 ⊢ (2nd ‘⟨{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}⟩}⟩) = {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}⟩}
68	63, 67	eqtri 2217	. . . . 5 ⊢ (2nd ‘𝐿) = {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}⟩}
69	61, 68	elrab2 2923	. . . 4 ⊢ ((𝑥 +Q 1Q) ∈ (2nd ‘𝐿) ↔ ((𝑥 +Q 1Q) ∈ Q ∧ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)}, {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞}⟩))
70	11, 54, 69	sylanbrc 417	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝑥 +Q 1Q) ∈ (2nd ‘𝐿))
71		eleq1 2259	. . . 4 ⊢ (𝑡 = (𝑥 +Q 1Q) → (𝑡 ∈ (2nd ‘𝐿) ↔ (𝑥 +Q 1Q) ∈ (2nd ‘𝐿)))
72	71	rspcev 2868	. . 3 ⊢ (((𝑥 +Q 1Q) ∈ Q ∧ (𝑥 +Q 1Q) ∈ (2nd ‘𝐿)) → ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿))
73	11, 70, 72	syl2anc 411	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿))
74	7, 73	rexlimddv 2619	1 ⊢ (𝜑 → ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∃wrex 2476  {crab 2479  ⟨cop 3626   class class class wbr 4034  ⟶wf 5255  ‘cfv 5259  (class class class)co 5925  1^st c1st 6205  2^nd c2nd 6206  1_oc1o 6476  [cec 6599  Ncnpi 7356   <_N clti 7359   ~_Q ceq 7363  Qcnq 7364  1_Qc1q 7365   +_Q cplq 7366  *_Qcrq 7368   <_Q cltq 7369  Pcnp 7375   +_P cpp 7377  <_P cltp 7379

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iplp 7552  df-iltp 7554

This theorem is referenced by:  caucvgprprlemm  7780