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Theorem limsupbnd2 15423

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 11254	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℝ^*
5	3, 4	sstrdi 3993	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
6		supxrunb1 13294	. . . . . . . . . 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 256	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℝ ∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗)
9		ifcl 4572	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ) → if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ)
10		breq1 5150	. . . . . . . . . 10 ⊢ (𝑛 = if(𝑘 ≤ 𝑚, 𝑚, 𝑘) → (𝑛 ≤ 𝑗 ↔ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗))
11	10	rexbidv 3178	. . . . . . . . 9 ⊢ (𝑛 = if(𝑘 ≤ 𝑚, 𝑚, 𝑘) → (∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗 ↔ ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗))
12	11	rspccva 3611	. . . . . . . 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 596	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗)
14		r19.29 3114	. . . . . . . 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 776	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑘 ∈ ℝ)
16		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝑚 ∈ ℝ)
17	16	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑚 ∈ ℝ)
18		max1 13160	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
19	15, 17, 18	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
20	17, 15, 9	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ)
21	3	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐵 ⊆ ℝ)
22	21	sselda 3981	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑗 ∈ ℝ)
23		letr 11304	. . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . . 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 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ (𝐹‘𝑗)))
28		max2 13162	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
29	15, 17, 28	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘))
30		letr 11304	. . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗 → 𝑚 ≤ 𝑗))
33	32	adantld 491	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚 ≤ 𝑗))
34		eqid 2732	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))
35	34	limsupgf 15415	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < )):ℝ⟶ℝ^*
36	35	ffvelcdmi 7082	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℝ → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))‘𝑚) ∈ ℝ^*)
37	36	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))‘𝑚) ∈ ℝ^*)
38	37	xrleidd 13127	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))‘𝑚))
39	38	adantrr 715	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 15417	. . . . . . . . . . . . . . 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 1376	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∀𝑗 ∈ 𝐵 (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))‘𝑚)))
46	45	r19.21bi 3248	. . . . . . . . . . . 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 513	. . . . . . . . . 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 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝐴 ∈ ℝ^*)
51	41	ffvelcdmda 7083	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (𝐹‘𝑗) ∈ ℝ^*)
52	42	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))‘𝑚) ∈ ℝ^*)
53		xrletr 13133	. . . . . . . . . . 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 1371	. . . . . . . . . 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 3155	. . . . . . . 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 692	. . . . . 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 468	. . . . 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 3155	. . . 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 3154	. . 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 15418	. . 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 1371	. 2 ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ^), ℝ^, < ))‘𝑚)))
65	62, 64	mpbird 256	1 ⊢ (𝜑 → 𝐴 ≤ (lim sup‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ∩ cin 3946 ⊆ wss 3947 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 “ cima 5678 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 supcsup 9431 ℝcr 11105 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 [,)cico 13322 lim supclsp 15410

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-ico 13326 df-limsup 15411

This theorem is referenced by: caucvgrlem 15615 limsupre 44343

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