Theorem caucvgprprlem1 8023

Description: Lemma for caucvgprpr 8026. 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 7638	. . . . . 6 ⊢ <N ⊆ (N × N)
7	6	brel 4801	. . . . 5 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
8	5, 7	syl 14	. . . 4 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
9	8	simprd 114	. . 3 ⊢ (𝜑 → 𝐾 ∈ N)
10	1, 9	ffvelcdmd 5812	. 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 7997	. . . . 5 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
14		nnnq 7736	. . . . . . . 8 ⊢ (𝐾 ∈ N → [⟨𝐾, 1o⟩] ~Q ∈ Q)
15	9, 14	syl 14	. . . . . . 7 ⊢ (𝜑 → [⟨𝐾, 1o⟩] ~Q ∈ Q)
16		recclnq 7706	. . . . . . 7 ⊢ ([⟨𝐾, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q)
17		nqprlu 7861	. . . . . . 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 7933	. . . . . 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 1274	. . . . 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 147	. . . 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 3882	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝐾 → ⟨𝑟, 1o⟩ = ⟨𝐾, 1o⟩)
23	22	eceq1d 6802	. . . . . . . . . . 11 ⊢ (𝑟 = 𝐾 → [⟨𝑟, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
24	23	fveq2d 5673	. . . . . . . . . 10 ⊢ (𝑟 = 𝐾 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
25	24	breq2d 4120	. . . . . . . . 9 ⊢ (𝑟 = 𝐾 → (𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
26	25	abbidv 2352	. . . . . . . 8 ⊢ (𝑟 = 𝐾 → {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )})
27	24	breq1d 4118	. . . . . . . . 9 ⊢ (𝑟 = 𝐾 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢))
28	27	abbidv 2352	. . . . . . . 8 ⊢ (𝑟 = 𝐾 → {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢})
29	26, 28	opeq12d 3890	. . . . . . 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 6065	. . . . . 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 5669	. . . . . . 7 ⊢ (𝑟 = 𝐾 → (𝐹‘𝑟) = (𝐹‘𝐾))
32	31	oveq1d 6064	. . . . . 6 ⊢ (𝑟 = 𝐾 → ((𝐹‘𝑟) +P 𝑄) = ((𝐹‘𝐾) +P 𝑄))
33	30, 32	breq12d 4121	. . . . 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 2920	. . . 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 411	. . 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 4111	. . . . . . . 8 ⊢ (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
37	36	cbvabv 2359	. . . . . . 7 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}
38		breq2 4112	. . . . . . . 8 ⊢ (𝑢 = 𝑞 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞))
39	38	cbvabv 2359	. . . . . . 7 ⊢ {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}
40	37, 39	opeq12i 3887	. . . . . 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 6060	. . . . 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 4115	. . . 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 2549	. . 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 122	. 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 8022	1 ⊢ (𝜑 → (𝐹‘𝐾)<P (𝐿 +P 𝑄))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  {cab 2218  ∀wral 2520  ∃wrex 2521  {crab 2524  ⟨cop 3691   class class class wbr 4108  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049  1_oc1o 6639  [cec 6764  Ncnpi 7586   <_N clti 7589   ~_Q ceq 7593  Qcnq 7594   +_Q cplq 7596  *_Qcrq 7598   <_Q cltq 7599  Pcnp 7605   +_P cpp 7607  <_P cltp 7609

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7618  df-pli 7619  df-mi 7620  df-lti 7621  df-plpq 7658  df-mpq 7659  df-enq 7661  df-nqqs 7662  df-plqqs 7663  df-mqqs 7664  df-1nqqs 7665  df-rq 7666  df-ltnqqs 7667  df-enq0 7738  df-nq0 7739  df-0nq0 7740  df-plq0 7741  df-mq0 7742  df-inp 7780  df-iplp 7782  df-iltp 7784

This theorem is referenced by:  caucvgprprlemlim  8025