Theorem caucvgprprlem1 6807

Description: Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-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}⟩}⟩
caucvgprprlemlim.q	⊢ (𝜑 → 𝑄 ∈ P)
caucvgprprlemlim.jk	⊢ (𝜑 → 𝐽 <N 𝐾)
caucvgprprlemlim.jkq	⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)

Assertion

Ref	Expression
caucvgprprlem1	⊢ (𝜑 → (𝐹‘𝐾)<P (𝐿 +P 𝑄))

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

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

Proof of Theorem caucvgprprlem1

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	. 2 ⊢ 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		caucvgprprlemlim.jk	. . . . 5 ⊢ (𝜑 → 𝐽 <N 𝐾)
6		ltrelpi 6422	. . . . . 6 ⊢ <N ⊆ (N × N)
7	6	brel 4392	. . . . 5 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
8	5, 7	syl 14	. . . 4 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
9	8	simprd 107	. . 3 ⊢ (𝜑 → 𝐾 ∈ N)
10	1, 9	ffvelrnd 5303	. 2 ⊢ (𝜑 → (𝐹‘𝐾) ∈ P)
11		caucvgprprlemlim.q	. 2 ⊢ (𝜑 → 𝑄 ∈ P)
12		caucvgprprlemlim.jkq	. . . . . 6 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
13	5, 12	caucvgprprlemk 6781	. . . . 5 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
14		nnnq 6520	. . . . . . . 8 ⊢ (𝐾 ∈ N → [⟨𝐾, 1𝑜⟩] ~Q ∈ Q)
15	9, 14	syl 14	. . . . . . 7 ⊢ (𝜑 → [⟨𝐾, 1𝑜⟩] ~Q ∈ Q)
16		recclnq 6490	. . . . . . 7 ⊢ ([⟨𝐾, 1𝑜⟩] ~Q ∈ Q → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
17		nqprlu 6645	. . . . . . 7 ⊢ ((*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
18	15, 16, 17	3syl 17	. . . . . 6 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
19		ltaprg 6717	. . . . . 6 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P ∧ 𝑄 ∈ P ∧ (𝐹‘𝐾) ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄 ↔ ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝐾) +P 𝑄)))
20	18, 11, 10, 19	syl3anc 1135	. . . . 5 ⊢ (𝜑 → (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄 ↔ ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝐾) +P 𝑄)))
21	13, 20	mpbid 135	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝐾) +P 𝑄))
22		opeq1 3549	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝐾 → ⟨𝑟, 1𝑜⟩ = ⟨𝐾, 1𝑜⟩)
23	22	eceq1d 6142	. . . . . . . . . . 11 ⊢ (𝑟 = 𝐾 → [⟨𝑟, 1𝑜⟩] ~Q = [⟨𝐾, 1𝑜⟩] ~Q )
24	23	fveq2d 5182	. . . . . . . . . 10 ⊢ (𝑟 = 𝐾 → (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))
25	24	breq2d 3776	. . . . . . . . 9 ⊢ (𝑟 = 𝐾 → (𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
26	25	abbidv 2155	. . . . . . . 8 ⊢ (𝑟 = 𝐾 → {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )})
27	24	breq1d 3774	. . . . . . . . 9 ⊢ (𝑟 = 𝐾 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢))
28	27	abbidv 2155	. . . . . . . 8 ⊢ (𝑟 = 𝐾 → {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢})
29	26, 28	opeq12d 3557	. . . . . . 7 ⊢ (𝑟 = 𝐾 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)
30	29	oveq2d 5528	. . . . . 6 ⊢ (𝑟 = 𝐾 → ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))
31		fveq2 5178	. . . . . . 7 ⊢ (𝑟 = 𝐾 → (𝐹‘𝑟) = (𝐹‘𝐾))
32	31	oveq1d 5527	. . . . . 6 ⊢ (𝑟 = 𝐾 → ((𝐹‘𝑟) +P 𝑄) = ((𝐹‘𝐾) +P 𝑄))
33	30, 32	breq12d 3777	. . . . 5 ⊢ (𝑟 = 𝐾 → (((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝑟) +P 𝑄) ↔ ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝐾) +P 𝑄)))
34	33	rspcev 2656	. . . 4 ⊢ ((𝐾 ∈ N ∧ ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝐾) +P 𝑄)) → ∃𝑟 ∈ N ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝑟) +P 𝑄))
35	9, 21, 34	syl2anc 391	. . 3 ⊢ (𝜑 → ∃𝑟 ∈ N ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝑟) +P 𝑄))
36		breq1 3767	. . . . . . . 8 ⊢ (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )))
37	36	cbvabv 2161	. . . . . . 7 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}
38		breq2 3768	. . . . . . . 8 ⊢ (𝑢 = 𝑞 → ((*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞))
39	38	cbvabv 2161	. . . . . . 7 ⊢ {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}
40	37, 39	opeq12i 3554	. . . . . 6 ⊢ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩
41	40	oveq2i 5523	. . . . 5 ⊢ ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)
42	41	breq1i 3771	. . . 4 ⊢ (((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝑟) +P 𝑄) ↔ ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹‘𝑟) +P 𝑄))
43	42	rexbii 2331	. . 3 ⊢ (∃𝑟 ∈ N ((𝐹‘𝐾) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ((𝐹‘𝑟) +P 𝑄) ↔ ∃𝑟 ∈ N ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹‘𝑟) +P 𝑄))
44	35, 43	sylib 127	. 2 ⊢ (𝜑 → ∃𝑟 ∈ N ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹‘𝑟) +P 𝑄))
45	1, 2, 3, 4, 10, 11, 44	caucvgprprlemaddq 6806	1 ⊢ (𝜑 → (𝐹‘𝐾)<P (𝐿 +P 𝑄))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  {crab 2310  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   <_N clti 6373   ~_Q ceq 6377  Qcnq 6378   +_Q cplq 6380  *_Qcrq 6382   <_Q cltq 6383  Pcnp 6389   +_P cpp 6391  <_P cltp 6393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-iltp 6568

This theorem is referenced by:  caucvgprprlemlim  6809