Theorem caucvgprprlemmu 7496

Description: Lemma for caucvgprpr 7513. 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 7116	. . . . 5 ⊢ 1o ∈ N
3	2	a1i 9	. . . 4 ⊢ (𝜑 → 1o ∈ N)
4	1, 3	ffvelrnd 5549	. . 3 ⊢ (𝜑 → (𝐹‘1o) ∈ P)
5		prop 7276	. . 3 ⊢ ((𝐹‘1o) ∈ P → ⟨(1st ‘(𝐹‘1o)), (2nd ‘(𝐹‘1o))⟩ ∈ P)
6		prmu 7279	. . 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 520	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → 𝑥 ∈ Q)
9		1nq 7167	. . . 4 ⊢ 1Q ∈ Q
10		addclnq 7176	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 1Q ∈ Q) → (𝑥 +Q 1Q) ∈ Q)
11	8, 9, 10	sylancl 409	. . 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 521	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → 𝑥 ∈ (2nd ‘(𝐹‘1o)))
14	4	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝐹‘1o) ∈ P)
15		nqpru 7353	. . . . . . . . 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 408	. . . . . . . 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 146	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝐹‘1o)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑥}, {𝑞 ∣ 𝑥 <Q 𝑞}⟩)
18		ltaprg 7420	. . . . . . . . 9 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
19	18	adantl 275	. . . . . . . 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 7348	. . . . . . . . 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 7348	. . . . . . . . 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 7379	. . . . . . . . 9 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
25	24	adantl 275	. . . . . . . 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 5933	. . . . . . 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 146	. . . . . 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 7152	. . . . . . . . . . . . 13 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
29	28	fveq2i 5417	. . . . . . . . . . . 12 ⊢ (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
30		rec1nq 7196	. . . . . . . . . . . 12 ⊢ (*Q‘1Q) = 1Q
31	29, 30	eqtr3i 2160	. . . . . . . . . . 11 ⊢ (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
32	31	breq2i 3932	. . . . . . . . . 10 ⊢ (𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q ) ↔ 𝑝 <Q 1Q)
33	32	abbii 2253	. . . . . . . . 9 ⊢ {𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q 1Q}
34	31	breq1i 3931	. . . . . . . . . 10 ⊢ ((*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞 ↔ 1Q <Q 𝑞)
35	34	abbii 2253	. . . . . . . . 9 ⊢ {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ 1Q <Q 𝑞}
36	33, 35	opeq12i 3705	. . . . . . . 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 5778	. . . . . . 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 7362	. . . . . . 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 409	. . . . . 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 3955	. . . . 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 5414	. . . . . . . 8 ⊢ (𝑟 = 1o → (𝐹‘𝑟) = (𝐹‘1o))
43		opeq1 3700	. . . . . . . . . . . . 13 ⊢ (𝑟 = 1o → ⟨𝑟, 1o⟩ = ⟨1o, 1o⟩)
44	43	eceq1d 6458	. . . . . . . . . . . 12 ⊢ (𝑟 = 1o → [⟨𝑟, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
45	44	fveq2d 5418	. . . . . . . . . . 11 ⊢ (𝑟 = 1o → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
46	45	breq2d 3936	. . . . . . . . . 10 ⊢ (𝑟 = 1o → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )))
47	46	abbidv 2255	. . . . . . . . 9 ⊢ (𝑟 = 1o → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )})
48	45	breq1d 3934	. . . . . . . . . 10 ⊢ (𝑟 = 1o → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞))
49	48	abbidv 2255	. . . . . . . . 9 ⊢ (𝑟 = 1o → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞})
50	47, 49	opeq12d 3708	. . . . . . . 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 5785	. . . . . . 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 3934	. . . . . 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 2784	. . . . 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 408	. . . 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 3928	. . . . . . . . 9 ⊢ (𝑢 = (𝑥 +Q 1Q) → (𝑝 <Q 𝑢 ↔ 𝑝 <Q (𝑥 +Q 1Q)))
56	55	abbidv 2255	. . . . . . . 8 ⊢ (𝑢 = (𝑥 +Q 1Q) → {𝑝 ∣ 𝑝 <Q 𝑢} = {𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)})
57		breq1 3927	. . . . . . . . 9 ⊢ (𝑢 = (𝑥 +Q 1Q) → (𝑢 <Q 𝑞 ↔ (𝑥 +Q 1Q) <Q 𝑞))
58	57	abbidv 2255	. . . . . . . 8 ⊢ (𝑢 = (𝑥 +Q 1Q) → {𝑞 ∣ 𝑢 <Q 𝑞} = {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞})
59	56, 58	opeq12d 3708	. . . . . . 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 3936	. . . . . 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 2436	. . . . 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 5417	. . . . . 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 7164	. . . . . . . 8 ⊢ Q ∈ V
65	64	rabex 4067	. . . . . . 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 4067	. . . . . . 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 6038	. . . . . 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 2158	. . . . 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 2838	. . . 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 413	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → (𝑥 +Q 1Q) ∈ (2nd ‘𝐿))
71		eleq1 2200	. . . 4 ⊢ (𝑡 = (𝑥 +Q 1Q) → (𝑡 ∈ (2nd ‘𝐿) ↔ (𝑥 +Q 1Q) ∈ (2nd ‘𝐿)))
72	71	rspcev 2784	. . 3 ⊢ (((𝑥 +Q 1Q) ∈ Q ∧ (𝑥 +Q 1Q) ∈ (2nd ‘𝐿)) → ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿))
73	11, 70, 72	syl2anc 408	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (2nd ‘(𝐹‘1o)))) → ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿))
74	7, 73	rexlimddv 2552	1 ⊢ (𝜑 → ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  {cab 2123  ∀wral 2414  ∃wrex 2415  {crab 2418  ⟨cop 3525   class class class wbr 3924  ⟶wf 5114  ‘cfv 5118  (class class class)co 5767  1^st c1st 6029  2^nd c2nd 6030  1_oc1o 6299  [cec 6420  Ncnpi 7073   <_N clti 7076   ~_Q ceq 7080  Qcnq 7081  1_Qc1q 7082   +_Q cplq 7083  *_Qcrq 7085   <_Q cltq 7086  Pcnp 7092   +_P cpp 7094  <_P cltp 7096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iplp 7269  df-iltp 7271

This theorem is referenced by:  caucvgprprlemm  7497