Theorem caucvgprprlemelu 8017

Description: Lemma for caucvgprpr 8043. 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 4118	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (𝑝 <Q 𝑢 ↔ 𝑝 <Q 𝑋))
2	1	abbidv 2354	. . . . . 6 ⊢ (𝑢 = 𝑋 → {𝑝 ∣ 𝑝 <Q 𝑢} = {𝑝 ∣ 𝑝 <Q 𝑋})
3		breq1 4117	. . . . . . 7 ⊢ (𝑢 = 𝑋 → (𝑢 <Q 𝑞 ↔ 𝑋 <Q 𝑞))
4	3	abbidv 2354	. . . . . 6 ⊢ (𝑢 = 𝑋 → {𝑞 ∣ 𝑢 <Q 𝑞} = {𝑞 ∣ 𝑋 <Q 𝑞})
5	2, 4	opeq12d 3896	. . . . 5 ⊢ (𝑢 = 𝑋 → ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩)
6	5	breq2d 4126	. . . 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 2545	. . 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 5678	. . . 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 7694	. . . . . 6 ⊢ Q ∈ V
11	10	rabex 4261	. . . . 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 4261	. . . . 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 6354	. . . 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 2255	. . 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 2979	. 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 5675	. . . . . . 7 ⊢ (𝑟 = 𝑎 → (𝐹‘𝑟) = (𝐹‘𝑎))
17		opeq1 3888	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑎 → ⟨𝑟, 1o⟩ = ⟨𝑎, 1o⟩)
18	17	eceq1d 6816	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑎 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
19	18	fveq2d 5679	. . . . . . . . . 10 ⊢ (𝑟 = 𝑎 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
20	19	breq2d 4126	. . . . . . . . 9 ⊢ (𝑟 = 𝑎 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
21	20	abbidv 2354	. . . . . . . 8 ⊢ (𝑟 = 𝑎 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )})
22	19	breq1d 4124	. . . . . . . . 9 ⊢ (𝑟 = 𝑎 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞))
23	22	abbidv 2354	. . . . . . . 8 ⊢ (𝑟 = 𝑎 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞})
24	21, 23	opeq12d 3896	. . . . . . 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 6076	. . . . . 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 4124	. . . . 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 2781	. . . 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 5675	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
29		opeq1 3888	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ⟨𝑎, 1o⟩ = ⟨𝑏, 1o⟩)
30	29	eceq1d 6816	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → [⟨𝑎, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
31	30	fveq2d 5679	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (*Q‘[⟨𝑎, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
32	31	breq2d 4126	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
33	32	abbidv 2354	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1o⟩] ~Q )})
34	31	breq1d 4124	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞))
35	34	abbidv 2354	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → {𝑞 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑏, 1o⟩] ~Q ) <Q 𝑞})
36	33, 35	opeq12d 3896	. . . . . . 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 6076	. . . . . 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 4124	. . . . 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 2781	. . . 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 1398   ∈ wcel 2205  {cab 2220  ∃wrex 2523  {crab 2526  ⟨cop 3697   class class class wbr 4114  ‘cfv 5357  (class class class)co 6058  2^nd c2nd 6346  1_oc1o 6653  [cec 6778  Ncnpi 7603   ~_Q ceq 7610  Qcnq 7611   +_Q cplq 7613  *_Qcrq 7615   <_Q cltq 7616   +_P cpp 7624  <_P cltp 7626

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 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  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-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-2nd 6348  df-ec 6782  df-qs 6786  df-ni 7635  df-nqqs 7679

This theorem is referenced by:  caucvgprprlemopu  8030  caucvgprprlemupu  8031  caucvgprprlemdisj  8033  caucvgprprlemloc  8034