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Theorem caucvgrlem 15021

Description: Lemma for caurcvgr 15022. (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 10617	. . . . . . . . 9 ⊢ ℝ ∈ V
4	3	ssex 5189	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V)
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
6	3	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
7		fex2 7620	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ V ∧ ℝ ∈ V) → 𝐹 ∈ V)
8	1, 5, 6, 7	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
9		limsupcl 14822	. . . . . 6 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ^*)
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
11	10	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (lim sup‘𝐹) ∈ ℝ^*)
12	1	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝐹:𝐴⟶ℝ)
13		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑗 ∈ 𝐴)
14	12, 13	ffvelrnd 6829	. . . . 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 12419	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ)
17	16	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑅 ∈ ℝ)
18	14, 17	readdcld 10659	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) + 𝑅) ∈ ℝ)
19		mnfxr 10687	. . . . . 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 11057	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) − 𝑅) ∈ ℝ)
22	21	rexrd 10680	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) − 𝑅) ∈ ℝ^*)
23	21	mnfltd 12507	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → -∞ < ((𝐹‘𝑗) − 𝑅))
24	2	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝐴 ⊆ ℝ)
25		ressxr 10674	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
26		fss 6501	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℝ^) → 𝐹:𝐴⟶ℝ^)
27	1, 25, 26	sylancl 589	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)
28	27	adantr 484	. . . . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → sup(𝐴, ℝ^*, < ) = +∞)
31	24, 13	sseldd 3916	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑗 ∈ ℝ)
32		simprr 772	. . . . . . . . 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 5034	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝑗 ≤ 𝑘 ↔ 𝑗 ≤ 𝑚))
34	33	imbrov2fvoveq 7160	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅) ↔ (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅)))
35	34	cbvralvw 3396	. . . . . . . . 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 221	. . . . . . . 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	ffvelrnda 6828	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → (𝐹‘𝑚) ∈ ℝ)
38	14	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ)
39	37, 38	resubcld 11057	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((𝐹‘𝑚) − (𝐹‘𝑗)) ∈ ℝ)
40	39	recnd 10658	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((𝐹‘𝑚) − (𝐹‘𝑗)) ∈ ℂ)
41	40	abscld 14788	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → 𝑅 ∈ ℝ)
43		ltle 10718	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 14786	. . . . . . . . . . . 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 242	. . . . . . . . . . 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 486	. . . . . . . . . . 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 3144	. . . . . . . 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 5033	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (𝑛 ≤ 𝑚 ↔ 𝑗 ≤ 𝑚))
53	52	rspceaimv 3576	. . . . . . 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 587	. . . . . 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 14832	. . . . 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 12542	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → -∞ < (lim sup‘𝐹))
57	18	rexrd 10680	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) + 𝑅) ∈ ℝ^*)
58	41	adantrr 716	. . . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑅 ∈ ℝ)
60		simprr 772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑗 ≤ 𝑚)
61		simplrr 777	. . . . . . . . . . . . 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 770	. . . . . . . . . . . . 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 3573	. . . . . . . . . . . 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 10777	. . . . . . . . . 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 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (𝐹‘𝑚) ∈ ℝ)
67	14	adantr 484	. . . . . . . . . . 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 14786	. . . . . . . . . 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 235	. . . . . . . . 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 499	. . . . . . . 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 460	. . . . . . 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 3149	. . . . . 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 3576	. . . . . 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 587	. . . . 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 14831	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (lim sup‘𝐹) ≤ ((𝐹‘𝑗) + 𝑅))
76		xrre 12550	. . . 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 837	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (lim sup‘𝐹) ∈ ℝ)
78	77	adantr 484	. . . . . . . . . 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 11057	. . . . . . . . 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 10658	. . . . . . . 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 14788	. . . . . . 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 11699	. . . . . . . 8 ⊢ 2 ∈ ℝ
83		remulcl 10611	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (2 · 𝑅) ∈ ℝ)
84	82, 59, 83	sylancr 590	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (2 · 𝑅) ∈ ℝ)
85		3re 11705	. . . . . . . 8 ⊢ 3 ∈ ℝ
86		remulcl 10611	. . . . . . . 8 ⊢ ((3 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (3 · 𝑅) ∈ ℝ)
87	85, 59, 86	sylancr 590	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (3 · 𝑅) ∈ ℝ)
88	66	recnd 10658	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (𝐹‘𝑚) ∈ ℂ)
89	78	recnd 10658	. . . . . . . . 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 14805	. . . . . . . 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 11057	. . . . . . . . . 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 484	. . . . . . . . . 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 10658	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑅 ∈ ℂ)
94	93	2timesd 11868	. . . . . . . . . . . . 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 7151	. . . . . . . . . . . 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 11017	. . . . . . . . . . . 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 2836	. . . . . . . . . . 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 11057	. . . . . . . . . . . 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 11226	. . . . . . . . . . . . 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 260	. . . . . . . . . . . 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 11245	. . . . . . . . . . 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 5052	. . . . . . . . . 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 484	. . . . . . . . . 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 10786	. . . . . . . . 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 484	. . . . . . . . . 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 10659	. . . . . . . . . 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 484	. . . . . . . . . 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 10659	. . . . . . . . . . . 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 11226	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 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 11243	. . . . . . . . . . 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 10652	. . . . . . . . . . . 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 7151	. . . . . . . . . . . 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 2836	. . . . . . . . . . 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 5056	. . . . . . . . . 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 10786	. . . . . . . . 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 14786	. . . . . . . . 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 712	. . . . . . . 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 5052	. . . . . . 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 11797	. . . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑅 ∈ ℝ⁺)
125	124	adantr 484	. . . . . . . . 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 12460	. . . . . . . 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 236	. . . . . . 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 10787	. . . . . 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 460	. . . . 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 3149	. . . 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 7160	. . . . 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 3396	. . . 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 237	. . 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 515	. 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 5034	. . . . 5 ⊢ (𝑥 = 𝑅 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))
136	135	imbi2d 344	. . . 4 ⊢ (𝑥 = 𝑅 → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅)))
137	136	rexralbidv 3260	. . 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 3573	. 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))
140	134, 139	reximddv 3234	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 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℝcr 10525 + caddc 10529 · cmul 10531 +∞cpnf 10661 -∞cmnf 10662 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 2c2 11680 3c3 11681 ℝ⁺crp 12377 abscabs 14585 lim supclsp 14819

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820

This theorem is referenced by: caurcvgr 15022

W3C validator