Theorem limsupbnd2 15418

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 11188	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℝ*
5	3, 4	sstrdi 3948	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ ℝ*)
6		supxrunb1 13246	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℝ ∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗)
9		ifcl 4527	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ) → if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ)
10		breq1 5103	. . . . . . . . . 10 ⊢ (𝑛 = if(𝑘 ≤ 𝑚, 𝑚, 𝑘) → (𝑛 ≤ 𝑗 ↔ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗))
11	10	rexbidv 3162	. . . . . . . . 9 ⊢ (𝑛 = if(𝑘 ≤ 𝑚, 𝑚, 𝑘) → (∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗 ↔ ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗))
12	11	rspccva 3577	. . . . . . . 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 597	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗)
14		r19.29 3101	. . . . . . . 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 778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑘 ∈ ℝ)
16		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝑚 ∈ ℝ)
17	16	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑚 ∈ ℝ)
18		max1 13112	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
19	15, 17, 18	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
20	17, 15, 9	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ)
21	3	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐵 ⊆ ℝ)
22	21	sselda 3935	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑗 ∈ ℝ)
23		letr 11239	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 696	. . . . . . . . . . . . 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 410	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ (𝐹‘𝑗)))
28		max2 13114	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
29	15, 17, 28	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
30		letr 11239	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 696	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗 → 𝑚 ≤ 𝑗))
33	32	adantld 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚 ≤ 𝑗))
34		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
35	34	limsupgf 15410	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
36	35	ffvelcdmi 7037	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℝ → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
37	36	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
38	37	xrleidd 13078	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
39	38	adantrr 718	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 15412	. . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∀𝑗 ∈ 𝐵 (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
46	45	r19.21bi 3230	. . . . . . . . . . . 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 512	. . . . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝐴 ∈ ℝ*)
51	41	ffvelcdmda 7038	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (𝐹‘𝑗) ∈ ℝ*)
52	42	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
53		xrletr 13084	. . . . . . . . . . 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 1374	. . . . . . . . . 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 3139	. . . . . . . 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 695	. . . . . 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 467	. . . . 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 3139	. . . 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 3138	. . 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 15413	. . 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 1374	. 2 ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
65	62, 64	mpbird 257	1 ⊢ (𝜑 → 𝐴 ≤ (lim sup‘𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3902   ⊆ wss 3903  ifcif 4481   class class class wbr 5100   ↦ cmpt 5181   “ cima 5635  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  supcsup 9355  ℝcr 11037  +∞cpnf 11175  ℝ^*cxr 11177   < clt 11178   ≤ cle 11179  [,)cico 13275  lim supclsp 15405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9357  df-inf 9358  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-ico 13279  df-limsup 15406

This theorem is referenced by:  caucvgrlem  15608  limsupre  45993