Theorem limsupbnd2 15560

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 11271	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℝ*
5	3, 4	sstrdi 3952	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ ℝ*)
6		supxrunb1 13363	. . . . . . . . . 10 ⊢ (𝐵 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
7	5, 6	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑛 ∈ ℝ ∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
8	2, 7	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℝ ∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗)
9		ifcl 4538	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ) → if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ)
10		breq1 5117	. . . . . . . . . 10 ⊢ (𝑛 = if(𝑘 ≤ 𝑚, 𝑚, 𝑘) → (𝑛 ≤ 𝑗 ↔ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗))
11	10	rexbidv 3192	. . . . . . . . 9 ⊢ (𝑛 = if(𝑘 ≤ 𝑚, 𝑚, 𝑘) → (∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗 ↔ ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗))
12	11	rspccva 3583	. . . . . . . 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 608	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗)
14		r19.29 3131	. . . . . . . 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 790	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑘 ∈ ℝ)
16		simprl 783	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝑚 ∈ ℝ)
17	16	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑚 ∈ ℝ)
18		max1 13229	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
19	15, 17, 18	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
20	17, 15, 9	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ)
21	3	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐵 ⊆ ℝ)
22	21	sselda 3940	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑗 ∈ ℝ)
23		letr 11322	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 708	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗 → 𝑘 ≤ 𝑗))
26	25	imim1d 83	. . . . . . . . . . . 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 416	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ (𝐹‘𝑗)))
28		max2 13231	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
29	15, 17, 28	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
30		letr 11322	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 708	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗 → 𝑚 ≤ 𝑗))
33	32	adantld 496	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚 ≤ 𝑗))
34		eqid 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
35	34	limsupgf 15552	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
36	35	ffvelcdmi 7085	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℝ → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
37	36	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
38	37	xrleidd 13195	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
39	38	adantrr 730	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐹:𝐵⟶ℝ*)
42	16, 36	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
43	34	limsupgle 15554	. . . . . . . . . . . . . . 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 1403	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∀𝑗 ∈ 𝐵 (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
46	45	r19.21bi 3260	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
47	33, 46	syld 48	. . . . . . . . . . 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 522	. . . . . . . . . 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 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝐴 ∈ ℝ*)
51	41	ffvelcdmda 7086	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (𝐹‘𝑗) ∈ ℝ*)
52	42	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
53		xrletr 13201	. . . . . . . . . . 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 1398	. . . . . . . . . 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 48	. . . . . . . . 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 3169	. . . . . . . 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 35	. . . . . . 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 707	. . . . . 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 473	. . . . 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 3169	. . . 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 3168	. . 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 16	. 2 ⊢ (𝜑 → ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
63	34	limsuple 15555	. . 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 1398	. 2 ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
65	62, 64	mpbird 260	1 ⊢ (𝜑 → 𝐴 ≤ (lim sup‘𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092   ∩ cin 3907   ⊆ wss 3908  ifcif 4492   class class class wbr 5114   ↦ cmpt 5197   “ cima 5669  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423  supcsup 9410  ℝcr 11117  +∞cpnf 11258  ℝ^*cxr 11260   < clt 11261   ≤ cle 11262  [,)cico 13392  lim supclsp 15547

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195  ax-pre-sup 11196

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-po 5574  df-so 5575  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-sup 9412  df-inf 9413  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-ico 13396  df-limsup 15548

This theorem is used by:  caucvgrlem  15750  limsupre  46396