Theorem caucvgprprlemclphr 7667

Description: Lemma for caucvgprpr 7674. The putative limit is a positive real. Like caucvgprprlemcl 7666 but without a disjoint variable condition between 𝜑 and 𝑟. (Contributed by Jim Kingdon, 19-Jun-2021.)

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
caucvgprprlemclphr	⊢ (𝜑 → 𝐿 ∈ P)

Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟   𝐹,𝑙,𝑢,𝑟,𝑘   𝑛,𝐹,𝑘   𝑘,𝐿   𝑢,𝑙,𝑝,𝑞,𝑟   𝑚,𝑟   𝑘,𝑝,𝑞,𝑟   𝑢,𝑛,𝑙,𝑘

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

Proof of Theorem caucvgprprlemclphr

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprpr.f	. 2 ⊢ (𝜑 → 𝐹:N⟶P)
2		caucvgprpr.cau	. 2 ⊢ (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}⟩))))
3		caucvgprpr.bnd	. 2 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))
4		caucvgprpr.lim	. . 3 ⊢ 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}⟩}⟩
5		opeq1 3765	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑠 → ⟨𝑟, 1o⟩ = ⟨𝑠, 1o⟩)
6	5	eceq1d 6549	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑠, 1o⟩] ~Q )
7	6	fveq2d 5500	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑠 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑠, 1o⟩] ~Q ))
8	7	oveq2d 5869	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )))
9	8	breq2d 4001	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))))
10	9	abbidv 2288	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → {𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))})
11	8	breq1d 3999	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞))
12	11	abbidv 2288	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞})
13	10, 12	opeq12d 3773	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩)
14		fveq2 5496	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝐹‘𝑟) = (𝐹‘𝑠))
15	13, 14	breq12d 4002	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑠)))
16	15	cbvrexv 2697	. . . . . 6 ⊢ (∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ∃𝑠 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑠))
17	16	a1i 9	. . . . 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) ↔ ∃𝑠 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑠)))
18	17	rabbiia 2715	. . . 4 ⊢ {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)} = {𝑙 ∈ Q ∣ ∃𝑠 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑠)}
19	7	breq2d 4001	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )))
20	19	abbidv 2288	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )})
21	7	breq1d 3999	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞))
22	21	abbidv 2288	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞})
23	20, 22	opeq12d 3773	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)
24	14, 23	oveq12d 5871	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘𝑠) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩))
25	24	breq1d 3999	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (((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‘𝑠) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩))
26	25	cbvrexv 2697	. . . . . 6 ⊢ (∃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}⟩ ↔ ∃𝑠 ∈ N ((𝐹‘𝑠) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩)
27	26	a1i 9	. . . . 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}⟩ ↔ ∃𝑠 ∈ N ((𝐹‘𝑠) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩))
28	27	rabbiia 2715	. . . 4 ⊢ {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 ∣ ∃𝑠 ∈ N ((𝐹‘𝑠) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}
29	18, 28	opeq12i 3770	. . 3 ⊢ ⟨{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}⟩}⟩ = ⟨{𝑙 ∈ Q ∣ ∃𝑠 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑠)}, {𝑢 ∈ Q ∣ ∃𝑠 ∈ N ((𝐹‘𝑠) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩
30	4, 29	eqtri 2191	. 2 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑠 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑠)}, {𝑢 ∈ Q ∣ ∃𝑠 ∈ N ((𝐹‘𝑠) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩
31	1, 2, 3, 30	caucvgprprlemcl 7666	1 ⊢ (𝜑 → 𝐿 ∈ P)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  {crab 2452  ⟨cop 3586   class class class wbr 3989  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  1_oc1o 6388  [cec 6511  Ncnpi 7234   <_N clti 7237   ~_Q ceq 7241  Qcnq 7242   +_Q cplq 7244  *_Qcrq 7246   <_Q cltq 7247  Pcnp 7253   +_P cpp 7255  <_P cltp 7257

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432

This theorem is referenced by:  caucvgprprlemexbt  7668  caucvgprprlemexb  7669