Theorem caucvgprprlemelu 7819

Description: Lemma for caucvgprpr 7845. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.)

Hypothesis

Ref	Expression
caucvgprprlemell.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
caucvgprprlemelu	⊢ (𝑋 ∈ (2nd ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩))

Distinct variable groups:   𝐹,𝑏   𝐹,𝑙,𝑟   𝑢,𝐹,𝑟   𝑋,𝑏,𝑝   𝑋,𝑙,𝑟,𝑝   𝑢,𝑋,𝑝   𝑋,𝑞,𝑏   𝑞,𝑙,𝑟   𝑢,𝑞

Allowed substitution hints:   𝐹(𝑞,𝑝)   𝐿(𝑢,𝑟,𝑞,𝑝,𝑏,𝑙)

Proof of Theorem caucvgprprlemelu

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4055	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (𝑝 <Q 𝑢 ↔ 𝑝 <Q 𝑋))
2	1	abbidv 2324	. . . . . 6 ⊢ (𝑢 = 𝑋 → {𝑝 ∣ 𝑝 <Q 𝑢} = {𝑝 ∣ 𝑝 <Q 𝑋})
3		breq1 4054	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (𝑢 <Q 𝑞 ↔ 𝑋 <Q 𝑞))
4	3	abbidv 2324	. . . . . 6 ⊢ (𝑢 = 𝑋 → {𝑞 ∣ 𝑢 <Q 𝑞} = {𝑞 ∣ 𝑋 <Q 𝑞})
5	2, 4	opeq12d 3833	. . . . 5 ⊢ (𝑢 = 𝑋 → ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩)
6	5	breq2d 4063	. . . 4 ⊢ (𝑢 = 𝑋 → (((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 ∣ 𝑋 <Q 𝑞}⟩))
7	6	rexbidv 2508	. . 3 ⊢ (𝑢 = 𝑋 → (∃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 ∣ 𝑋 <Q 𝑞}⟩))
8		caucvgprprlemell.lim	. . . . 5 ⊢ 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}⟩}⟩
9	8	fveq2i 5592	. . . 4 ⊢ (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}⟩}⟩)
10		nqex 7496	. . . . . 6 ⊢ Q ∈ V
11	10	rabex 4196	. . . . 5 ⊢ {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
12	10	rabex 4196	. . . . 5 ⊢ {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
13	11, 12	op2nd 6246	. . . 4 ⊢ (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}⟩}
14	9, 13	eqtri 2227	. . 3 ⊢ (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}⟩}
15	7, 14	elrab2 2936	. 2 ⊢ (𝑋 ∈ (2nd ‘𝐿) ↔ (𝑋 ∈ 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 ∣ 𝑋 <Q 𝑞}⟩))
16		fveq2 5589	. . . . . . 7 ⊢ (𝑟 = 𝑎 → (𝐹‘𝑟) = (𝐹‘𝑎))
17		opeq1 3825	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑎 → ⟨𝑟, 1o⟩ = ⟨𝑎, 1o⟩)
18	17	eceq1d 6669	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑎 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
19	18	fveq2d 5593	. . . . . . . . . 10 ⊢ (𝑟 = 𝑎 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
20	19	breq2d 4063	. . . . . . . . 9 ⊢ (𝑟 = 𝑎 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
21	20	abbidv 2324	. . . . . . . 8 ⊢ (𝑟 = 𝑎 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )})
22	19	breq1d 4061	. . . . . . . . 9 ⊢ (𝑟 = 𝑎 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞))
23	22	abbidv 2324	. . . . . . . 8 ⊢ (𝑟 = 𝑎 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞})
24	21, 23	opeq12d 3833	. . . . . . 7 ⊢ (𝑟 = 𝑎 → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩)
25	16, 24	oveq12d 5975	. . . . . 6 ⊢ (𝑟 = 𝑎 → ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘𝑎) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩))
26	25	breq1d 4061	. . . . 5 ⊢ (𝑟 = 𝑎 → (((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩ ↔ ((𝐹‘𝑎) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩))
27	26	cbvrexv 2740	. . . 4 ⊢ (∃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 ∣ 𝑋 <Q 𝑞}⟩ ↔ ∃𝑎 ∈ N ((𝐹‘𝑎) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩)
28		fveq2 5589	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
29		opeq1 3825	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ⟨𝑎, 1o⟩ = ⟨𝑏, 1o⟩)
30	29	eceq1d 6669	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → [⟨𝑎, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
31	30	fveq2d 5593	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (*Q‘[⟨𝑎, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
32	31	breq2d 4063	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
33	32	abbidv 2324	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )})
34	31	breq1d 4061	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞))
35	34	abbidv 2324	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞})
36	33, 35	opeq12d 3833	. . . . . . 7 ⊢ (𝑎 = 𝑏 → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)
37	28, 36	oveq12d 5975	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝐹‘𝑎) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩))
38	37	breq1d 4061	. . . . 5 ⊢ (𝑎 = 𝑏 → (((𝐹‘𝑎) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩ ↔ ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩))
39	38	cbvrexv 2740	. . . 4 ⊢ (∃𝑎 ∈ N ((𝐹‘𝑎) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩ ↔ ∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩)
40	27, 39	bitri 184	. . 3 ⊢ (∃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 ∣ 𝑋 <Q 𝑞}⟩ ↔ ∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩)
41	40	anbi2i 457	. 2 ⊢ ((𝑋 ∈ 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 ∣ 𝑋 <Q 𝑞}⟩) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩))
42	15, 41	bitri 184	1 ⊢ (𝑋 ∈ (2nd ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2177  {cab 2192  ∃wrex 2486  {crab 2489  ⟨cop 3641   class class class wbr 4051  ‘cfv 5280  (class class class)co 5957  2^nd c2nd 6238  1_oc1o 6508  [cec 6631  Ncnpi 7405   ~_Q ceq 7412  Qcnq 7413   +_Q cplq 7415  *_Qcrq 7417   <_Q cltq 7418   +_P cpp 7426  <_P cltp 7428

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-2nd 6240  df-ec 6635  df-qs 6639  df-ni 7437  df-nqqs 7481

This theorem is referenced by:  caucvgprprlemopu  7832  caucvgprprlemupu  7833  caucvgprprlemdisj  7835  caucvgprprlemloc  7836