Theorem limsupbnd2 15493

Description: If a sequence is eventually greater than 𝐴, then the limsup is also greater than 𝐴. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)

Hypotheses

Ref	Expression
limsupbnd.1	⊢ (𝜑 → 𝐵 ⊆ ℝ)
limsupbnd.2	⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)
limsupbnd.3	⊢ (𝜑 → 𝐴 ∈ ℝ*)
limsupbnd2.4	⊢ (𝜑 → sup(𝐵, ℝ*, < ) = +∞)
limsupbnd2.5	⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)))

Assertion

Ref	Expression
limsupbnd2	⊢ (𝜑 → 𝐴 ≤ (lim sup‘𝐹))

Distinct variable groups:   𝑗,𝑘,𝐴   𝐵,𝑗,𝑘   𝑗,𝐹,𝑘   𝜑,𝑗,𝑘

Proof of Theorem limsupbnd2

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupbnd2.5	. . 3 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)))
2		limsupbnd2.4	. . . . . . . . 9 ⊢ (𝜑 → sup(𝐵, ℝ*, < ) = +∞)
3		limsupbnd.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
4		ressxr 11223	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℝ*
5	3, 4	sstrdi 3948	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ ℝ*)
6		supxrunb1 13319	. . . . . . . . . 10 ⊢ (𝐵 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
7	5, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑛 ∈ ℝ ∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
8	2, 7	mpbird 259	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℝ ∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗)
9		ifcl 4525	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ) → if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ)
10		breq1 5102	. . . . . . . . . 10 ⊢ (𝑛 = if(𝑘 ≤ 𝑚, 𝑚, 𝑘) → (𝑛 ≤ 𝑗 ↔ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗))
11	10	rexbidv 3185	. . . . . . . . 9 ⊢ (𝑛 = if(𝑘 ≤ 𝑚, 𝑚, 𝑘) → (∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗 ↔ ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗))
12	11	rspccva 3580	. . . . . . . 8 ⊢ ((∀𝑛 ∈ ℝ ∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗 ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ) → ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗)
13	8, 9, 12	syl2an 605	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗)
14		r19.29 3124	. . . . . . . 8 ⊢ ((∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → ∃𝑗 ∈ 𝐵 ((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗))
15		simplrr 787	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑘 ∈ ℝ)
16		simprl 780	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝑚 ∈ ℝ)
17	16	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑚 ∈ ℝ)
18		max1 13185	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
19	15, 17, 18	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
20	17, 15, 9	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ)
21	3	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐵 ⊆ ℝ)
22	21	sselda 3936	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑗 ∈ ℝ)
23		letr 11274	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ ∧ 𝑗 ∈ ℝ) → ((𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑘 ≤ 𝑗))
24	15, 20, 22, 23	syl3anc 1389	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑘 ≤ 𝑗))
25	19, 24	mpand 705	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗 → 𝑘 ≤ 𝑗))
26	25	imim1d 82	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) → (if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗))))
27	26	impd 414	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ (𝐹‘𝑗)))
28		max2 13187	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
29	15, 17, 28	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
30		letr 11274	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℝ ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ ∧ 𝑗 ∈ ℝ) → ((𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚 ≤ 𝑗))
31	17, 20, 22, 30	syl3anc 1389	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚 ≤ 𝑗))
32	29, 31	mpand 705	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗 → 𝑚 ≤ 𝑗))
33	32	adantld 494	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚 ≤ 𝑗))
34		eqid 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
35	34	limsupgf 15485	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
36	35	ffvelcdmi 7060	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℝ → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
37	36	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
38	37	xrleidd 13151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
39	38	adantrr 727	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
40		limsupbnd.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)
41	40	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐹:𝐵⟶ℝ*)
42	16, 36	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
43	34	limsupgle 15487	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*) ∧ 𝑚 ∈ ℝ ∧ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ↔ ∀𝑗 ∈ 𝐵 (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))))
44	21, 41, 16, 42, 43	syl211anc 1394	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ↔ ∀𝑗 ∈ 𝐵 (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))))
45	39, 44	mpbid 234	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∀𝑗 ∈ 𝐵 (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
46	45	r19.21bi 3253	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
47	33, 46	syld 47	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
48	27, 47	jcad 520	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → (𝐴 ≤ (𝐹‘𝑗) ∧ (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))))
49		limsupbnd.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
50	49	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝐴 ∈ ℝ*)
51	41	ffvelcdmda 7061	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (𝐹‘𝑗) ∈ ℝ*)
52	42	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
53		xrletr 13157	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐹‘𝑗) ∈ ℝ* ∧ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*) → ((𝐴 ≤ (𝐹‘𝑗) ∧ (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
54	50, 51, 52, 53	syl3anc 1389	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝐴 ≤ (𝐹‘𝑗) ∧ (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
55	48, 54	syld 47	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
56	55	rexlimdva 3162	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (∃𝑗 ∈ 𝐵 ((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
57	14, 56	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ((∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
58	13, 57	mpan2d 704	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
59	58	anassrs 471	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
60	59	rexlimdva 3162	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
61	60	ralrimdva 3161	. . 3 ⊢ (𝜑 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) → ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
62	1, 61	mpd 15	. 2 ⊢ (𝜑 → ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
63	34	limsuple 15488	. . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
64	3, 40, 49, 63	syl3anc 1389	. 2 ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
65	62, 64	mpbird 259	1 ⊢ (𝜑 → 𝐴 ≤ (lim sup‘𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   ∩ cin 3903   ⊆ wss 3904  ifcif 4479   class class class wbr 5099   ↦ cmpt 5180   “ cima 5648  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  supcsup 9383  ℝcr 11069  +∞cpnf 11210  ℝ^*cxr 11212   < clt 11213   ≤ cle 11214  [,)cico 13348  lim supclsp 15480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-po 5553  df-so 5554  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-sup 9385  df-inf 9386  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-ico 13352  df-limsup 15481

This theorem is referenced by:  caucvgrlem  15683  limsupre  46179