Theorem caucvgprprlem1 7517

Description: Lemma for caucvgprpr 7520. 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 7132	. . . . . 6 ⊢ <N ⊆ (N × N)
7	6	brel 4591	. . . . 5 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
8	5, 7	syl 14	. . . 4 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
9	8	simprd 113	. . 3 ⊢ (𝜑 → 𝐾 ∈ N)
10	1, 9	ffvelrnd 5556	. 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 7491	. . . . 5 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
14		nnnq 7230	. . . . . . . 8 ⊢ (𝐾 ∈ N → [⟨𝐾, 1o⟩] ~Q ∈ Q)
15	9, 14	syl 14	. . . . . . 7 ⊢ (𝜑 → [⟨𝐾, 1o⟩] ~Q ∈ Q)
16		recclnq 7200	. . . . . . 7 ⊢ ([⟨𝐾, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q)
17		nqprlu 7355	. . . . . . 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 7427	. . . . . 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 1216	. . . . 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 146	. . . 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 3705	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝐾 → ⟨𝑟, 1o⟩ = ⟨𝐾, 1o⟩)
23	22	eceq1d 6465	. . . . . . . . . . 11 ⊢ (𝑟 = 𝐾 → [⟨𝑟, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
24	23	fveq2d 5425	. . . . . . . . . 10 ⊢ (𝑟 = 𝐾 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
25	24	breq2d 3941	. . . . . . . . 9 ⊢ (𝑟 = 𝐾 → (𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
26	25	abbidv 2257	. . . . . . . 8 ⊢ (𝑟 = 𝐾 → {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )})
27	24	breq1d 3939	. . . . . . . . 9 ⊢ (𝑟 = 𝐾 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢))
28	27	abbidv 2257	. . . . . . . 8 ⊢ (𝑟 = 𝐾 → {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢})
29	26, 28	opeq12d 3713	. . . . . . 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 5790	. . . . . 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 5421	. . . . . . 7 ⊢ (𝑟 = 𝐾 → (𝐹‘𝑟) = (𝐹‘𝐾))
32	31	oveq1d 5789	. . . . . 6 ⊢ (𝑟 = 𝐾 → ((𝐹‘𝑟) +P 𝑄) = ((𝐹‘𝐾) +P 𝑄))
33	30, 32	breq12d 3942	. . . . 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 2789	. . . 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 408	. . 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 3932	. . . . . . . 8 ⊢ (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
37	36	cbvabv 2264	. . . . . . 7 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}
38		breq2 3933	. . . . . . . 8 ⊢ (𝑢 = 𝑞 → ((*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞))
39	38	cbvabv 2264	. . . . . . 7 ⊢ {𝑢 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}
40	37, 39	opeq12i 3710	. . . . . 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 5785	. . . . 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 3936	. . . 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 2442	. . 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 121	. 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 7516	1 ⊢ (𝜑 → (𝐹‘𝐾)<P (𝐿 +P 𝑄))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∃wrex 2417  {crab 2420  ⟨cop 3530   class class class wbr 3929  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  1_oc1o 6306  [cec 6427  Ncnpi 7080   <_N clti 7083   ~_Q ceq 7087  Qcnq 7088   +_Q cplq 7090  *_Qcrq 7092   <_Q cltq 7093  Pcnp 7099   +_P cpp 7101  <_P cltp 7103

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-iltp 7278

This theorem is referenced by:  caucvgprprlemlim  7519