Theorem caucvgprprlemclphr 7968

Description: Lemma for caucvgprpr 7975. The putative limit is a positive real. Like caucvgprprlemcl 7967 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 3867	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑠 → ⟨𝑟, 1o⟩ = ⟨𝑠, 1o⟩)
6	5	eceq1d 6781	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑠, 1o⟩] ~Q )
7	6	fveq2d 5652	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑠 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑠, 1o⟩] ~Q ))
8	7	oveq2d 6044	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )))
9	8	breq2d 4105	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))))
10	9	abbidv 2350	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → {𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q ))})
11	8	breq1d 4103	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑙 +Q (*Q‘[⟨𝑠, 1o⟩] ~Q )) <Q 𝑞))
12	11	abbidv 2350	. . . . . . . . 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 3875	. . . . . . . 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 5648	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (𝐹‘𝑟) = (𝐹‘𝑠))
15	13, 14	breq12d 4106	. . . . . . 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 2769	. . . . . 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 2789	. . . 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 4105	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → (𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )))
20	19	abbidv 2350	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑠, 1o⟩] ~Q )})
21	7	breq1d 4103	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞))
22	21	abbidv 2350	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞} = {𝑞 ∣ (*Q‘[⟨𝑠, 1o⟩] ~Q ) <Q 𝑞})
23	20, 22	opeq12d 3875	. . . . . . . . 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 6046	. . . . . . . 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 4103	. . . . . . 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 2769	. . . . . 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 2789	. . . 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 3872	. . 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 2252	. 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 7967	1 ⊢ (𝜑 → 𝐿 ∈ P)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  {crab 2515  ⟨cop 3676   class class class wbr 4093  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  1_oc1o 6618  [cec 6743  Ncnpi 7535   <_N clti 7538   ~_Q ceq 7542  Qcnq 7543   +_Q cplq 7545  *_Qcrq 7547   <_Q cltq 7548  Pcnp 7554   +_P cpp 7556  <_P cltp 7558

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-iplp 7731  df-iltp 7733

This theorem is referenced by:  caucvgprprlemexbt  7969  caucvgprprlemexb  7970