Theorem liminfval2 41459

Description: The superior limit, relativized to an unbounded set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
liminfval2.1	⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
liminfval2.2	⊢ (𝜑 → 𝐹 ∈ 𝑉)
liminfval2.3	⊢ (𝜑 → 𝐴 ⊆ ℝ)
liminfval2.4	⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)

Assertion

Ref	Expression
liminfval2	⊢ (𝜑 → (lim inf‘𝐹) = sup((𝐺 “ 𝐴), ℝ*, < ))

Distinct variable group:   𝑘,𝐹

Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐺(𝑘)   𝑉(𝑘)

Proof of Theorem liminfval2

Dummy variables 𝑛 𝑥 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		liminfval2.2	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		liminfval2.1	. . . . 5 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))
3		oveq1 6981	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑘[,)+∞) = (𝑗[,)+∞))
4	3	imaeq2d 5768	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐹 “ (𝑘[,)+∞)) = (𝐹 “ (𝑗[,)+∞)))
5	4	ineq1d 4074	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*))
6	5	infeq1d 8732	. . . . . 6 ⊢ (𝑘 = 𝑗 → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
7	6	cbvmptv 5026	. . . . 5 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
8	2, 7	eqtri 2799	. . . 4 ⊢ 𝐺 = (𝑗 ∈ ℝ ↦ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))
9	8	liminfval 41450	. . 3 ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
10	1, 9	syl 17	. 2 ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < ))
11		liminfval2.4	. . . . . . 7 ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)
12		liminfval2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
13	12	ssrexr 41116	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ*)
14		supxrunb1 12525	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 ↔ sup(𝐴, ℝ*, < ) = +∞))
16	11, 15	mpbird 249	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥)
17	8	liminfgf 41449	. . . . . . . . . . 11 ⊢ 𝐺:ℝ⟶ℝ*
18	17	ffvelrni 6673	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) ∈ ℝ*)
19	18	ad2antlr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ∈ ℝ*)
20		simpll 754	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝜑)
21		simprl 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ 𝐴)
22	12	sselda 3857	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
23	17	ffvelrni 6673	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) ∈ ℝ*)
24	22, 23	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ℝ*)
25	20, 21, 24	syl2anc 576	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ∈ ℝ*)
26		imassrn 5779	. . . . . . . . . . . 12 ⊢ (𝐺 “ 𝐴) ⊆ ran 𝐺
27		frn 6348	. . . . . . . . . . . . 13 ⊢ (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*)
28	17, 27	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran 𝐺 ⊆ ℝ*
29	26, 28	sstri 3866	. . . . . . . . . . 11 ⊢ (𝐺 “ 𝐴) ⊆ ℝ*
30		supxrcl 12521	. . . . . . . . . . 11 ⊢ ((𝐺 “ 𝐴) ⊆ ℝ* → sup((𝐺 “ 𝐴), ℝ*, < ) ∈ ℝ*)
31	29, 30	ax-mp 5	. . . . . . . . . 10 ⊢ sup((𝐺 “ 𝐴), ℝ*, < ) ∈ ℝ*
32	31	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → sup((𝐺 “ 𝐴), ℝ*, < ) ∈ ℝ*)
33		simplr 756	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ)
34	20, 21, 22	syl2anc 576	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ)
35		simprr 760	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ≤ 𝑥)
36		liminfgord 41445	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑛 ≤ 𝑥) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
37	33, 34, 35, 36	syl3anc 1351	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
38	8	liminfgval 41453	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ → (𝐺‘𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
39	38	ad2antlr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑛) = inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
40	8	liminfgval 41453	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (𝐺‘𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
41	22, 40	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
42	41	adantlr 702	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < ))
43	39, 42	breq12d 4940	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑛) ≤ (𝐺‘𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
44	43	adantrr 704	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → ((𝐺‘𝑛) ≤ (𝐺‘𝑥) ↔ inf(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝑥[,)+∞)) ∩ ℝ*), ℝ*, < )))
45	37, 44	mpbird 249	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ≤ (𝐺‘𝑥))
46	29	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺 “ 𝐴) ⊆ ℝ*)
47		nfv 1873	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝜑
48		inss2 4092	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ⊆ ℝ*
49		infxrcl 12539	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ⊆ ℝ* → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*)
50	48, 49	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*
51	50	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ ℝ*)
52	47, 51, 8	fnmptd 6317	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 Fn ℝ)
53	52	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺 Fn ℝ)
54		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
55	53, 22, 54	fnfvimad 6819	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
56		supxrub 12530	. . . . . . . . . . 11 ⊢ (((𝐺 “ 𝐴) ⊆ ℝ* ∧ (𝐺‘𝑥) ∈ (𝐺 “ 𝐴)) → (𝐺‘𝑥) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
57	46, 55, 56	syl2anc 576	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
58	20, 21, 57	syl2anc 576	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑥) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
59	19, 25, 32, 45, 58	xrletrd 12369	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑛 ≤ 𝑥)) → (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
60	59	rexlimdvaa 3227	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ) → (∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
61	60	ralimdva 3124	. . . . . 6 ⊢ (𝜑 → (∀𝑛 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑛 ≤ 𝑥 → ∀𝑛 ∈ ℝ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
62	16, 61	mpd 15	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℝ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
63		xrltso 12348	. . . . . . . . 9 ⊢ < Or ℝ*
64	63	infex 8748	. . . . . . . 8 ⊢ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
65	64	rgenw 3097	. . . . . . 7 ⊢ ∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V
66	8	fnmpt 6316	. . . . . . 7 ⊢ (∀𝑗 ∈ ℝ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V → 𝐺 Fn ℝ)
67	65, 66	ax-mp 5	. . . . . 6 ⊢ 𝐺 Fn ℝ
68		breq1 4930	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑛) → (𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
69	68	ralrn 6677	. . . . . 6 ⊢ (𝐺 Fn ℝ → (∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ ∀𝑛 ∈ ℝ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
70	67, 69	ax-mp 5	. . . . 5 ⊢ (∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ ∀𝑛 ∈ ℝ (𝐺‘𝑛) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
71	62, 70	sylibr 226	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
72		supxrleub 12532	. . . . 5 ⊢ ((ran 𝐺 ⊆ ℝ* ∧ sup((𝐺 “ 𝐴), ℝ*, < ) ∈ ℝ*) → (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < )))
73	28, 31, 72	mp2an 679	. . . 4 ⊢ (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝑥 ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
74	71, 73	sylibr 226	. . 3 ⊢ (𝜑 → sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺 “ 𝐴), ℝ*, < ))
75	26	a1i 11	. . . 4 ⊢ (𝜑 → (𝐺 “ 𝐴) ⊆ ran 𝐺)
76	28	a1i 11	. . . 4 ⊢ (𝜑 → ran 𝐺 ⊆ ℝ*)
77		supxrss 12538	. . . 4 ⊢ (((𝐺 “ 𝐴) ⊆ ran 𝐺 ∧ ran 𝐺 ⊆ ℝ*) → sup((𝐺 “ 𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
78	75, 76, 77	syl2anc 576	. . 3 ⊢ (𝜑 → sup((𝐺 “ 𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))
79		supxrcl 12521	. . . . 5 ⊢ (ran 𝐺 ⊆ ℝ* → sup(ran 𝐺, ℝ*, < ) ∈ ℝ*)
80	28, 79	ax-mp 5	. . . 4 ⊢ sup(ran 𝐺, ℝ*, < ) ∈ ℝ*
81		xrletri3 12361	. . . 4 ⊢ ((sup(ran 𝐺, ℝ*, < ) ∈ ℝ* ∧ sup((𝐺 “ 𝐴), ℝ*, < ) ∈ ℝ*) → (sup(ran 𝐺, ℝ*, < ) = sup((𝐺 “ 𝐴), ℝ*, < ) ↔ (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ∧ sup((𝐺 “ 𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < ))))
82	80, 31, 81	mp2an 679	. . 3 ⊢ (sup(ran 𝐺, ℝ*, < ) = sup((𝐺 “ 𝐴), ℝ*, < ) ↔ (sup(ran 𝐺, ℝ*, < ) ≤ sup((𝐺 “ 𝐴), ℝ*, < ) ∧ sup((𝐺 “ 𝐴), ℝ*, < ) ≤ sup(ran 𝐺, ℝ*, < )))
83	74, 78, 82	sylanbrc 575	. 2 ⊢ (𝜑 → sup(ran 𝐺, ℝ*, < ) = sup((𝐺 “ 𝐴), ℝ*, < ))
84	10, 83	eqtrd 2811	1 ⊢ (𝜑 → (lim inf‘𝐹) = sup((𝐺 “ 𝐴), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2048  ∀wral 3085  ∃wrex 3086  Vcvv 3412   ∩ cin 3827   ⊆ wss 3828   class class class wbr 4927   ↦ cmpt 5006  ran crn 5405   “ cima 5407   Fn wfn 6181  ⟶wf 6182  ‘cfv 6186  (class class class)co 6974  supcsup 8695  infcinf 8696  ℝcr 10330  +∞cpnf 10467  ℝ^*cxr 10469   < clt 10470   ≤ cle 10471  [,)cico 12553  lim infclsi 41442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277  ax-cnex 10387  ax-resscn 10388  ax-1cn 10389  ax-icn 10390  ax-addcl 10391  ax-addrcl 10392  ax-mulcl 10393  ax-mulrcl 10394  ax-mulcom 10395  ax-addass 10396  ax-mulass 10397  ax-distr 10398  ax-i2m1 10399  ax-1ne0 10400  ax-1rid 10401  ax-rnegex 10402  ax-rrecex 10403  ax-cnre 10404  ax-pre-lttri 10405  ax-pre-lttrn 10406  ax-pre-ltadd 10407  ax-pre-mulgt0 10408  ax-pre-sup 10409

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ne 2965  df-nel 3071  df-ral 3090  df-rex 3091  df-reu 3092  df-rmo 3093  df-rab 3094  df-v 3414  df-sbc 3681  df-csb 3786  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-op 4446  df-uni 4711  df-iun 4792  df-br 4928  df-opab 4990  df-mpt 5007  df-id 5309  df-po 5323  df-so 5324  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7498  df-2nd 7499  df-er 8085  df-en 8303  df-dom 8304  df-sdom 8305  df-sup 8697  df-inf 8698  df-pnf 10472  df-mnf 10473  df-xr 10474  df-ltxr 10475  df-le 10476  df-sub 10668  df-neg 10669  df-ico 12557  df-liminf 41443

This theorem is referenced by:  liminfresico  41462