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Theorem caucvgrlem 15635

Description: Lemma for caurcvgr 15636. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by AV, 12-Sep-2020.)

**Hypotheses**
Ref	Expression
caurcvgr.1	⊢ (𝜑 → 𝐴 ⊆ ℝ)
caurcvgr.2	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
caurcvgr.3	⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)
caurcvgr.4	⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
caucvgrlem.4	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)

**Assertion**
Ref	Expression
caucvgrlem	⊢ (𝜑 → ∃𝑗 ∈ 𝐴 ((lim sup‘𝐹) ∈ ℝ ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 𝑅))))

Distinct variable groups: 𝑗,𝑘,𝑥,𝐴 𝑗,𝐹,𝑘,𝑥 𝜑,𝑗,𝑘,𝑥 𝑅,𝑗,𝑘,𝑥

Proof of Theorem caucvgrlem Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caurcvgr.2	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
2		caurcvgr.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		reex 11129	. . . . . . . . 9 ⊢ ℝ ∈ V
4	3	ssex 5262	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V)
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
6	3	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
7		fex2 7887	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ V ∧ ℝ ∈ V) → 𝐹 ∈ V)
8	1, 5, 6, 7	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
9		limsupcl 15435	. . . . . 6 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ^*)
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (lim sup‘𝐹) ∈ ℝ^*)
12	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝐹:𝐴⟶ℝ)
13		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑗 ∈ 𝐴)
14	12, 13	ffvelcdmd 7037	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (𝐹‘𝑗) ∈ ℝ)
15		caucvgrlem.4	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
16	15	rpred 12986	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ)
17	16	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑅 ∈ ℝ)
18	14, 17	readdcld 11174	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) + 𝑅) ∈ ℝ)
19		mnfxr 11202	. . . . . 6 ⊢ -∞ ∈ ℝ^*
20	19	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → -∞ ∈ ℝ^*)
21	14, 17	resubcld 11578	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) − 𝑅) ∈ ℝ)
22	21	rexrd 11195	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) − 𝑅) ∈ ℝ^*)
23	21	mnfltd 13075	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → -∞ < ((𝐹‘𝑗) − 𝑅))
24	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝐴 ⊆ ℝ)
25		ressxr 11189	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
26		fss 6684	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℝ^) → 𝐹:𝐴⟶ℝ^)
27	1, 25, 26	sylancl 587	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)
28	27	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝐹:𝐴⟶ℝ^*)
29		caurcvgr.3	. . . . . . 7 ⊢ (𝜑 → sup(𝐴, ℝ^*, < ) = +∞)
30	29	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → sup(𝐴, ℝ^*, < ) = +∞)
31	24, 13	sseldd 3922	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑗 ∈ ℝ)
32		simprr 773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))
33		breq2 5089	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝑗 ≤ 𝑘 ↔ 𝑗 ≤ 𝑚))
34	33	imbrov2fvoveq 7392	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅) ↔ (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅)))
35	34	cbvralvw 3215	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅) ↔ ∀𝑚 ∈ 𝐴 (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅))
36	32, 35	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ∀𝑚 ∈ 𝐴 (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅))
37	12	ffvelcdmda 7036	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → (𝐹‘𝑚) ∈ ℝ)
38	14	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ)
39	37, 38	resubcld 11578	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((𝐹‘𝑚) − (𝐹‘𝑗)) ∈ ℝ)
40	39	recnd 11173	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((𝐹‘𝑚) − (𝐹‘𝑗)) ∈ ℂ)
41	40	abscld 15401	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) ∈ ℝ)
42	17	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → 𝑅 ∈ ℝ)
43		ltle 11234	. . . . . . . . . . . . 13 ⊢ (((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) ≤ 𝑅))
44	41, 42, 43	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) ≤ 𝑅))
45	37, 38, 42	absdifled 15399	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) ≤ 𝑅 ↔ (((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚) ∧ (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅))))
46	44, 45	sylibd 239	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅 → (((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚) ∧ (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅))))
47		simpl 482	. . . . . . . . . . 11 ⊢ ((((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚) ∧ (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅)) → ((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚))
48	46, 47	syl6 35	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅 → ((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚)))
49	48	imim2d 57	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅) → (𝑗 ≤ 𝑚 → ((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚))))
50	49	ralimdva 3149	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (∀𝑚 ∈ 𝐴 (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅) → ∀𝑚 ∈ 𝐴 (𝑗 ≤ 𝑚 → ((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚))))
51	36, 50	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ∀𝑚 ∈ 𝐴 (𝑗 ≤ 𝑚 → ((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚)))
52		breq1 5088	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (𝑛 ≤ 𝑚 ↔ 𝑗 ≤ 𝑚))
53	52	rspceaimv 3570	. . . . . . 7 ⊢ ((𝑗 ∈ ℝ ∧ ∀𝑚 ∈ 𝐴 (𝑗 ≤ 𝑚 → ((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚))) → ∃𝑛 ∈ ℝ ∀𝑚 ∈ 𝐴 (𝑛 ≤ 𝑚 → ((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚)))
54	31, 51, 53	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ∃𝑛 ∈ ℝ ∀𝑚 ∈ 𝐴 (𝑛 ≤ 𝑚 → ((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚)))
55	24, 28, 22, 30, 54	limsupbnd2 15445	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) − 𝑅) ≤ (lim sup‘𝐹))
56	20, 22, 11, 23, 55	xrltletrd 13112	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → -∞ < (lim sup‘𝐹))
57	18	rexrd 11195	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) + 𝑅) ∈ ℝ^*)
58	41	adantrr 718	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) ∈ ℝ)
59	17	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑅 ∈ ℝ)
60		simprr 773	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑗 ≤ 𝑚)
61		simplrr 778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))
62		simprl 771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑚 ∈ 𝐴)
63	34, 61, 62	rspcdva 3565	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅))
64	60, 63	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅)
65	58, 59, 64	ltled 11294	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) ≤ 𝑅)
66	37	adantrr 718	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (𝐹‘𝑚) ∈ ℝ)
67	14	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (𝐹‘𝑗) ∈ ℝ)
68	66, 67, 59	absdifled 15399	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) ≤ 𝑅 ↔ (((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚) ∧ (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅))))
69	65, 68	mpbid 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚) ∧ (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅)))
70	69	simprd 495	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅))
71	70	expr 456	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → (𝑗 ≤ 𝑚 → (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅)))
72	71	ralrimiva 3129	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ∀𝑚 ∈ 𝐴 (𝑗 ≤ 𝑚 → (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅)))
73	52	rspceaimv 3570	. . . . . 6 ⊢ ((𝑗 ∈ ℝ ∧ ∀𝑚 ∈ 𝐴 (𝑗 ≤ 𝑚 → (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅))) → ∃𝑛 ∈ ℝ ∀𝑚 ∈ 𝐴 (𝑛 ≤ 𝑚 → (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅)))
74	31, 72, 73	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ∃𝑛 ∈ ℝ ∀𝑚 ∈ 𝐴 (𝑛 ≤ 𝑚 → (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅)))
75	24, 28, 57, 74	limsupbnd1 15444	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (lim sup‘𝐹) ≤ ((𝐹‘𝑗) + 𝑅))
76		xrre 13121	. . . 4 ⊢ ((((lim sup‘𝐹) ∈ ℝ^* ∧ ((𝐹‘𝑗) + 𝑅) ∈ ℝ) ∧ (-∞ < (lim sup‘𝐹) ∧ (lim sup‘𝐹) ≤ ((𝐹‘𝑗) + 𝑅))) → (lim sup‘𝐹) ∈ ℝ)
77	11, 18, 56, 75, 76	syl22anc 839	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (lim sup‘𝐹) ∈ ℝ)
78	77	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (lim sup‘𝐹) ∈ ℝ)
79	66, 78	resubcld 11578	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) − (lim sup‘𝐹)) ∈ ℝ)
80	79	recnd 11173	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) − (lim sup‘𝐹)) ∈ ℂ)
81	80	abscld 15401	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (abs‘((𝐹‘𝑚) − (lim sup‘𝐹))) ∈ ℝ)
82		2re 12255	. . . . . . . 8 ⊢ 2 ∈ ℝ
83		remulcl 11123	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (2 · 𝑅) ∈ ℝ)
84	82, 59, 83	sylancr 588	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (2 · 𝑅) ∈ ℝ)
85		3re 12261	. . . . . . . 8 ⊢ 3 ∈ ℝ
86		remulcl 11123	. . . . . . . 8 ⊢ ((3 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (3 · 𝑅) ∈ ℝ)
87	85, 59, 86	sylancr 588	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (3 · 𝑅) ∈ ℝ)
88	66	recnd 11173	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (𝐹‘𝑚) ∈ ℂ)
89	78	recnd 11173	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (lim sup‘𝐹) ∈ ℂ)
90	88, 89	abssubd 15418	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (abs‘((𝐹‘𝑚) − (lim sup‘𝐹))) = (abs‘((lim sup‘𝐹) − (𝐹‘𝑚))))
91	66, 84	resubcld 11578	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) − (2 · 𝑅)) ∈ ℝ)
92	21	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑗) − 𝑅) ∈ ℝ)
93	59	recnd 11173	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑅 ∈ ℂ)
94	93	2timesd 12420	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (2 · 𝑅) = (𝑅 + 𝑅))
95	94	oveq2d 7383	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) − (2 · 𝑅)) = ((𝐹‘𝑚) − (𝑅 + 𝑅)))
96	88, 93, 93	subsub4d 11536	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (((𝐹‘𝑚) − 𝑅) − 𝑅) = ((𝐹‘𝑚) − (𝑅 + 𝑅)))
97	95, 96	eqtr4d 2774	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) − (2 · 𝑅)) = (((𝐹‘𝑚) − 𝑅) − 𝑅))
98	66, 59	resubcld 11578	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) − 𝑅) ∈ ℝ)
99	66, 59, 67	lesubaddd 11747	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (((𝐹‘𝑚) − 𝑅) ≤ (𝐹‘𝑗) ↔ (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅)))
100	70, 99	mpbird 257	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) − 𝑅) ≤ (𝐹‘𝑗))
101	98, 67, 59, 100	lesub1dd 11766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (((𝐹‘𝑚) − 𝑅) − 𝑅) ≤ ((𝐹‘𝑗) − 𝑅))
102	97, 101	eqbrtrd 5107	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) − (2 · 𝑅)) ≤ ((𝐹‘𝑗) − 𝑅))
103	55	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑗) − 𝑅) ≤ (lim sup‘𝐹))
104	91, 92, 78, 102, 103	letrd 11303	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) − (2 · 𝑅)) ≤ (lim sup‘𝐹))
105	18	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑗) + 𝑅) ∈ ℝ)
106	66, 84	readdcld 11174	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) + (2 · 𝑅)) ∈ ℝ)
107	75	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (lim sup‘𝐹) ≤ ((𝐹‘𝑗) + 𝑅))
108	66, 59	readdcld 11174	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) + 𝑅) ∈ ℝ)
109	69, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚))
110	67, 59, 66	lesubaddd 11747	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚) ↔ (𝐹‘𝑗) ≤ ((𝐹‘𝑚) + 𝑅)))
111	109, 110	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (𝐹‘𝑗) ≤ ((𝐹‘𝑚) + 𝑅))
112	67, 108, 59, 111	leadd1dd 11764	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑗) + 𝑅) ≤ (((𝐹‘𝑚) + 𝑅) + 𝑅))
113	88, 93, 93	addassd 11167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (((𝐹‘𝑚) + 𝑅) + 𝑅) = ((𝐹‘𝑚) + (𝑅 + 𝑅)))
114	94	oveq2d 7383	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑚) + (2 · 𝑅)) = ((𝐹‘𝑚) + (𝑅 + 𝑅)))
115	113, 114	eqtr4d 2774	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (((𝐹‘𝑚) + 𝑅) + 𝑅) = ((𝐹‘𝑚) + (2 · 𝑅)))
116	112, 115	breqtrd 5111	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((𝐹‘𝑗) + 𝑅) ≤ ((𝐹‘𝑚) + (2 · 𝑅)))
117	78, 105, 106, 107, 116	letrd 11303	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (lim sup‘𝐹) ≤ ((𝐹‘𝑚) + (2 · 𝑅)))
118	78, 66, 84	absdifled 15399	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → ((abs‘((lim sup‘𝐹) − (𝐹‘𝑚))) ≤ (2 · 𝑅) ↔ (((𝐹‘𝑚) − (2 · 𝑅)) ≤ (lim sup‘𝐹) ∧ (lim sup‘𝐹) ≤ ((𝐹‘𝑚) + (2 · 𝑅)))))
119	104, 117, 118	mpbir2and 714	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (abs‘((lim sup‘𝐹) − (𝐹‘𝑚))) ≤ (2 · 𝑅))
120	90, 119	eqbrtrd 5107	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (abs‘((𝐹‘𝑚) − (lim sup‘𝐹))) ≤ (2 · 𝑅))
121		2lt3 12348	. . . . . . . 8 ⊢ 2 < 3
122	82	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 2 ∈ ℝ)
123	85	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 3 ∈ ℝ)
124	15	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑅 ∈ ℝ⁺)
125	124	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑅 ∈ ℝ⁺)
126	122, 123, 125	ltmul1d 13027	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (2 < 3 ↔ (2 · 𝑅) < (3 · 𝑅)))
127	121, 126	mpbii 233	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (2 · 𝑅) < (3 · 𝑅))
128	81, 84, 87, 120, 127	lelttrd 11304	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (abs‘((𝐹‘𝑚) − (lim sup‘𝐹))) < (3 · 𝑅))
129	128	expr 456	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (lim sup‘𝐹))) < (3 · 𝑅)))
130	129	ralrimiva 3129	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ∀𝑚 ∈ 𝐴 (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (lim sup‘𝐹))) < (3 · 𝑅)))
131	33	imbrov2fvoveq 7392	. . . . 5 ⊢ (𝑘 = 𝑚 → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 𝑅)) ↔ (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (lim sup‘𝐹))) < (3 · 𝑅))))
132	131	cbvralvw 3215	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 𝑅)) ↔ ∀𝑚 ∈ 𝐴 (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (lim sup‘𝐹))) < (3 · 𝑅)))
133	130, 132	sylibr 234	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 𝑅)))
134	77, 133	jca 511	. 2 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((lim sup‘𝐹) ∈ ℝ ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 𝑅))))
135		breq2 5089	. . . . 5 ⊢ (𝑥 = 𝑅 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))
136	135	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑅 → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅)))
137	136	rexralbidv 3203	. . 3 ⊢ (𝑥 = 𝑅 → (∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅)))
138		caurcvgr.4	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
139	137, 138, 15	rspcdva 3565	. 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))
140	134, 139	reximddv 3153	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝐴 ((lim sup‘𝐹) ∈ ℝ ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 𝑅))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ∃wrex 3061 Vcvv 3429 ⊆ wss 3889 class class class wbr 5085 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 supcsup 9353 ℝcr 11037 + caddc 11041 · cmul 11043 +∞cpnf 11176 -∞cmnf 11177 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 − cmin 11377 2c2 12236 3c3 12237 ℝ⁺crp 12942 abscabs 15196 lim supclsp 15432

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 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-ico 13304 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433

This theorem is referenced by: caurcvgr 15636

W3C validator