Theorem caucvgprprlemmu 7890

Description: Lemma for caucvgprpr 7907. 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 7510	. . . . 5 ⊢ 1o ∈ N
3	2	a1i 9	. . . 4 ⊢ (𝜑 → 1o ∈ N)
4	1, 3	ffvelcdmd 5773	. . 3 ⊢ (𝜑 → (𝐹‘1o) ∈ P)
5		prop 7670	. . 3 ⊢ ((𝐹‘1o) ∈ P → ⟨(1st ‘(𝐹‘1o)), (2nd ‘(𝐹‘1o))⟩ ∈ P)
6		prmu 7673	. . 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 7561	. . . 4 ⊢ 1Q ∈ Q
10		addclnq 7570	. . . 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 7747	. . . . . . . . 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 7814	. . . . . . . . 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 7742	. . . . . . . . 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 7742	. . . . . . . . 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 7773	. . . . . . . . 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 6181	. . . . . . 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 7546	. . . . . . . . . . . . 13 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
29	28	fveq2i 5632	. . . . . . . . . . . 12 ⊢ (*Q‘1Q) = (*Q‘[⟨1o, 1o⟩] ~Q )
30		rec1nq 7590	. . . . . . . . . . . 12 ⊢ (*Q‘1Q) = 1Q
31	29, 30	eqtr3i 2252	. . . . . . . . . . 11 ⊢ (*Q‘[⟨1o, 1o⟩] ~Q ) = 1Q
32	31	breq2i 4091	. . . . . . . . . 10 ⊢ (𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q ) ↔ 𝑝 <Q 1Q)
33	32	abbii 2345	. . . . . . . . 9 ⊢ {𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q 1Q}
34	31	breq1i 4090	. . . . . . . . . 10 ⊢ ((*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞 ↔ 1Q <Q 𝑞)
35	34	abbii 2345	. . . . . . . . 9 ⊢ {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ 1Q <Q 𝑞}
36	33, 35	opeq12i 3862	. . . . . . . 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 6018	. . . . . . 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 7756	. . . . . . 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 4115	. . . . 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 5629	. . . . . . . 8 ⊢ (𝑟 = 1o → (𝐹‘𝑟) = (𝐹‘1o))
43		opeq1 3857	. . . . . . . . . . . . 13 ⊢ (𝑟 = 1o → ⟨𝑟, 1o⟩ = ⟨1o, 1o⟩)
44	43	eceq1d 6724	. . . . . . . . . . . 12 ⊢ (𝑟 = 1o → [⟨𝑟, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
45	44	fveq2d 5633	. . . . . . . . . . 11 ⊢ (𝑟 = 1o → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨1o, 1o⟩] ~Q ))
46	45	breq2d 4095	. . . . . . . . . 10 ⊢ (𝑟 = 1o → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )))
47	46	abbidv 2347	. . . . . . . . 9 ⊢ (𝑟 = 1o → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨1o, 1o⟩] ~Q )})
48	45	breq1d 4093	. . . . . . . . . 10 ⊢ (𝑟 = 1o → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞))
49	48	abbidv 2347	. . . . . . . . 9 ⊢ (𝑟 = 1o → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨1o, 1o⟩] ~Q ) <Q 𝑞})
50	47, 49	opeq12d 3865	. . . . . . . 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 6025	. . . . . . 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 4093	. . . . . 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 2907	. . . . 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 4087	. . . . . . . . 9 ⊢ (𝑢 = (𝑥 +Q 1Q) → (𝑝 <Q 𝑢 ↔ 𝑝 <Q (𝑥 +Q 1Q)))
56	55	abbidv 2347	. . . . . . . 8 ⊢ (𝑢 = (𝑥 +Q 1Q) → {𝑝 ∣ 𝑝 <Q 𝑢} = {𝑝 ∣ 𝑝 <Q (𝑥 +Q 1Q)})
57		breq1 4086	. . . . . . . . 9 ⊢ (𝑢 = (𝑥 +Q 1Q) → (𝑢 <Q 𝑞 ↔ (𝑥 +Q 1Q) <Q 𝑞))
58	57	abbidv 2347	. . . . . . . 8 ⊢ (𝑢 = (𝑥 +Q 1Q) → {𝑞 ∣ 𝑢 <Q 𝑞} = {𝑞 ∣ (𝑥 +Q 1Q) <Q 𝑞})
59	56, 58	opeq12d 3865	. . . . . . 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 4095	. . . . . 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 2531	. . . . 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 5632	. . . . . 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 7558	. . . . . . . 8 ⊢ Q ∈ V
65	64	rabex 4228	. . . . . . 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 4228	. . . . . . 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 6299	. . . . . 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 2250	. . . . 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 2962	. . . 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 2292	. . . 4 ⊢ (𝑡 = (𝑥 +Q 1Q) → (𝑡 ∈ (2nd ‘𝐿) ↔ (𝑥 +Q 1Q) ∈ (2nd ‘𝐿)))
72	71	rspcev 2907	. . 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 2653	1 ⊢ (𝜑 → ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  {crab 2512  ⟨cop 3669   class class class wbr 4083  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007  1^st c1st 6290  2^nd c2nd 6291  1_oc1o 6561  [cec 6686  Ncnpi 7467   <_N clti 7470   ~_Q ceq 7474  Qcnq 7475  1_Qc1q 7476   +_Q cplq 7477  *_Qcrq 7479   <_Q cltq 7480  Pcnp 7486   +_P cpp 7488  <_P cltp 7490

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7499  df-pli 7500  df-mi 7501  df-lti 7502  df-plpq 7539  df-mpq 7540  df-enq 7542  df-nqqs 7543  df-plqqs 7544  df-mqqs 7545  df-1nqqs 7546  df-rq 7547  df-ltnqqs 7548  df-enq0 7619  df-nq0 7620  df-0nq0 7621  df-plq0 7622  df-mq0 7623  df-inp 7661  df-iplp 7663  df-iltp 7665

This theorem is referenced by:  caucvgprprlemm  7891