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Theorem caucvgrlem 15690

Description: Lemma for caurcvgr 15691. (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 11157	. . . . . . . . 9 ⊢ ℝ ∈ V
4	3	ssex 5274	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V)
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
6	3	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
7		fex2 7911	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ V ∧ ℝ ∈ V) → 𝐹 ∈ V)
8	1, 5, 6, 7	syl3anc 1389	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
9		limsupcl 15490	. . . . . 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 780	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑗 ∈ 𝐴)
14	12, 13	ffvelcdmd 7060	. . . . 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 13030	. . . . . 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 11204	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) + 𝑅) ∈ ℝ)
19		mnfxr 11232	. . . . . 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 11608	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) − 𝑅) ∈ ℝ)
22	21	rexrd 11225	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) − 𝑅) ∈ ℝ^*)
23	21	mnfltd 13119	. . . . 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 11219	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
26		fss 6702	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℝ^) → 𝐹:𝐴⟶ℝ^)
27	1, 25, 26	sylancl 595	. . . . . . 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 3935	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑗 ∈ ℝ)
32		simprr 782	. . . . . . . . 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 5101	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝑗 ≤ 𝑘 ↔ 𝑗 ≤ 𝑚))
34	33	imbrov2fvoveq 7415	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅) ↔ (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅)))
35	34	cbvralvw 3239	. . . . . . . . 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 220	. . . . . . . 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 7059	. . . . . . . . . . . . . . . 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 11608	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((𝐹‘𝑚) − (𝐹‘𝑗)) ∈ ℝ)
40	39	recnd 11203	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((𝐹‘𝑚) − (𝐹‘𝑗)) ∈ ℂ)
41	40	abscld 15456	. . . . . . . . . . . . 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 11264	. . . . . . . . . . . . 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 593	. . . . . . . . . . . 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 15454	. . . . . . . . . . . 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 241	. . . . . . . . . . 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 3173	. . . . . . . 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 5100	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (𝑛 ≤ 𝑚 ↔ 𝑗 ≤ 𝑚))
53	52	rspceaimv 3586	. . . . . . 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 593	. . . . . 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 15500	. . . . 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 13156	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → -∞ < (lim sup‘𝐹))
57	18	rexrd 11225	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) + 𝑅) ∈ ℝ^*)
58	41	adantrr 727	. . . . . . . . . . 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 782	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑗 ≤ 𝑚)
61		simplrr 787	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 3581	. . . . . . . . . . . 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 11324	. . . . . . . . . 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 727	. . . . . . . . . . 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 15454	. . . . . . . . . 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 234	. . . . . . . . 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 3153	. . . . . 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 3586	. . . . . 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 593	. . . . 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 15499	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (lim sup‘𝐹) ≤ ((𝐹‘𝑗) + 𝑅))
76		xrre 13165	. . . 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 849	. . 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 11608	. . . . . . . . 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 11203	. . . . . . . 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 15456	. . . . . . 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 12285	. . . . . . . 8 ⊢ 2 ∈ ℝ
83		remulcl 11151	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (2 · 𝑅) ∈ ℝ)
84	82, 59, 83	sylancr 596	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (2 · 𝑅) ∈ ℝ)
85		3re 12291	. . . . . . . 8 ⊢ 3 ∈ ℝ
86		remulcl 11151	. . . . . . . 8 ⊢ ((3 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (3 · 𝑅) ∈ ℝ)
87	85, 59, 86	sylancr 596	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (3 · 𝑅) ∈ ℝ)
88	66	recnd 11203	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (𝐹‘𝑚) ∈ ℂ)
89	78	recnd 11203	. . . . . . . . 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 15473	. . . . . . . 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 11608	. . . . . . . . . 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 11203	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑅 ∈ ℂ)
94	93	2timesd 12457	. . . . . . . . . . . . 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 7406	. . . . . . . . . . . 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 11566	. . . . . . . . . . . 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 2799	. . . . . . . . . . 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 11608	. . . . . . . . . . . 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 11777	. . . . . . . . . . . . 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 259	. . . . . . . . . . . 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 11796	. . . . . . . . . . 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 5119	. . . . . . . . . 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 11333	. . . . . . . . 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 11204	. . . . . . . . . 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 11204	. . . . . . . . . . . 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 11777	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 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 11794	. . . . . . . . . . 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 11197	. . . . . . . . . . . 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 7406	. . . . . . . . . . . 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 2799	. . . . . . . . . . 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 5123	. . . . . . . . . 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 11333	. . . . . . . . 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 15454	. . . . . . . . 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 723	. . . . . . . 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 5119	. . . . . . 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 12384	. . . . . . . 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 13071	. . . . . . . 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 235	. . . . . . 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 11334	. . . . . 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 3153	. . . 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 7415	. . . . 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 3239	. . . 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 236	. . 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 519	. 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 5101	. . . . 5 ⊢ (𝑥 = 𝑅 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))
136	135	imbi2d 342	. . . 4 ⊢ (𝑥 = 𝑅 → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅)))
137	136	rexralbidv 3227	. . 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 3581	. 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))
140	134, 139	reximddv 3177	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 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 Vcvv 3453 ⊆ wss 3902 class class class wbr 5097 ⟶wf 6511 ‘cfv 6515 (class class class)co 7390 supcsup 9379 ℝcr 11065 + caddc 11069 · cmul 11071 +∞cpnf 11206 -∞cmnf 11207 ℝ^*cxr 11208 < clt 11209 ≤ cle 11210 − cmin 11407 2c2 12265 3c3 12266 ℝ⁺crp 12986 abscabs 15251 lim supclsp 15487

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-er 8671 df-en 8921 df-dom 8922 df-sdom 8923 df-sup 9381 df-inf 9382 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-n0 12475 df-z 12562 df-uz 12833 df-rp 12987 df-ico 13348 df-seq 14008 df-exp 14068 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15488

This theorem is referenced by: caurcvgr 15691

W3C validator