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Theorem caucvgrlem 15743

Description: Lemma for caurcvgr 15744. (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 11202	. . . . . . . . 9 ⊢ ℝ ∈ V
4	3	ssex 5293	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V)
5	2, 4	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
6	3	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
7		fex2 7935	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ V ∧ ℝ ∈ V) → 𝐹 ∈ V)
8	1, 5, 6, 7	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
9		limsupcl 15543	. . . . . 6 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ^*)
10	8, 9	syl 18	. . . . 5 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (lim sup‘𝐹) ∈ ℝ^*)
12	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝐹:𝐴⟶ℝ)
13		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑗 ∈ 𝐴)
14	12, 13	ffvelcdmd 7084	. . . . 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 13071	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ)
17	16	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑅 ∈ ℝ)
18	14, 17	readdcld 11249	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) + 𝑅) ∈ ℝ)
19		mnfxr 11277	. . . . . 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 11653	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) − 𝑅) ∈ ℝ)
22	21	rexrd 11270	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) − 𝑅) ∈ ℝ^*)
23	21	mnfltd 13160	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → -∞ < ((𝐹‘𝑗) − 𝑅))
24	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝐴 ⊆ ℝ)
25		ressxr 11264	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
26		fss 6726	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℝ^) → 𝐹:𝐴⟶ℝ^)
27	1, 25, 26	sylancl 598	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)
28	27	adantr 486	. . . . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → sup(𝐴, ℝ^*, < ) = +∞)
31	24, 13	sseldd 3939	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑗 ∈ ℝ)
32		simprr 785	. . . . . . . . 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 5115	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝑗 ≤ 𝑘 ↔ 𝑗 ≤ 𝑚))
34	33	imbrov2fvoveq 7441	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅) ↔ (𝑗 ≤ 𝑚 → (abs‘((𝐹‘𝑚) − (𝐹‘𝑗))) < 𝑅)))
35	34	cbvralvw 3245	. . . . . . . . 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	ffvelcdmda 7083	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → (𝐹‘𝑚) ∈ ℝ)
38	14	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ)
39	37, 38	resubcld 11653	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((𝐹‘𝑚) − (𝐹‘𝑗)) ∈ ℝ)
40	39	recnd 11248	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → ((𝐹‘𝑚) − (𝐹‘𝑗)) ∈ ℂ)
41	40	abscld 15509	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ 𝑚 ∈ 𝐴) → 𝑅 ∈ ℝ)
43		ltle 11309	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 15507	. . . . . . . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚) ∧ (𝐹‘𝑚) ≤ ((𝐹‘𝑗) + 𝑅)) → ((𝐹‘𝑗) − 𝑅) ≤ (𝐹‘𝑚))
48	46, 47	syl6 36	. . . . . . . . . 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 58	. . . . . . . . 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 3179	. . . . . . . 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 16	. . . . . . 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 5114	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (𝑛 ≤ 𝑚 ↔ 𝑗 ≤ 𝑚))
53	52	rspceaimv 3589	. . . . . . 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 596	. . . . . 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 15553	. . . . 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 13197	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → -∞ < (lim sup‘𝐹))
57	18	rexrd 11270	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → ((𝐹‘𝑗) + 𝑅) ∈ ℝ^*)
58	41	adantrr 730	. . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑅 ∈ ℝ)
60		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑗 ≤ 𝑚)
61		simplrr 790	. . . . . . . . . . . . 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 783	. . . . . . . . . . . . 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 3584	. . . . . . . . . . . 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 16	. . . . . . . . . . 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 11369	. . . . . . . . . 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 730	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (𝐹‘𝑚) ∈ ℝ)
67	14	adantr 486	. . . . . . . . . . 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 15507	. . . . . . . . . 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 501	. . . . . . . 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 462	. . . . . . 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 3159	. . . . . 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 3589	. . . . . 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 596	. . . . 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 15552	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (lim sup‘𝐹) ≤ ((𝐹‘𝑗) + 𝑅))
76		xrre 13206	. . . 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 852	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → (lim sup‘𝐹) ∈ ℝ)
78	77	adantr 486	. . . . . . . . . 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 11653	. . . . . . . . 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 11248	. . . . . . . 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 15509	. . . . . . 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 12326	. . . . . . . 8 ⊢ 2 ∈ ℝ
83		remulcl 11196	. . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (2 · 𝑅) ∈ ℝ)
84	82, 59, 83	sylancr 599	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (2 · 𝑅) ∈ ℝ)
85		3re 12332	. . . . . . . 8 ⊢ 3 ∈ ℝ
86		remulcl 11196	. . . . . . . 8 ⊢ ((3 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (3 · 𝑅) ∈ ℝ)
87	85, 59, 86	sylancr 599	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (3 · 𝑅) ∈ ℝ)
88	66	recnd 11248	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → (𝐹‘𝑚) ∈ ℂ)
89	78	recnd 11248	. . . . . . . . 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 15526	. . . . . . . 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 11653	. . . . . . . . . 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 486	. . . . . . . . . 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 11248	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) ∧ (𝑚 ∈ 𝐴 ∧ 𝑗 ≤ 𝑚)) → 𝑅 ∈ ℂ)
94	93	2timesd 12498	. . . . . . . . . . . . 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 7432	. . . . . . . . . . . 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 11611	. . . . . . . . . . . 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 2803	. . . . . . . . . . 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 11653	. . . . . . . . . . . 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 11822	. . . . . . . . . . . . 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 11841	. . . . . . . . . . 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 5135	. . . . . . . . . 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 486	. . . . . . . . . 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 11378	. . . . . . . . 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 486	. . . . . . . . . 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 11249	. . . . . . . . . 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 486	. . . . . . . . . 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 11249	. . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 11822	. . . . . . . . . . . . 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 11839	. . . . . . . . . . 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 11242	. . . . . . . . . . . 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 7432	. . . . . . . . . . . 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 2803	. . . . . . . . . . 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 5139	. . . . . . . . . 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 11378	. . . . . . . . 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 15507	. . . . . . . . 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 726	. . . . . . . 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 5135	. . . . . . 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 12425	. . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))) → 𝑅 ∈ ℝ⁺)
125	124	adantr 486	. . . . . . . . 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 13112	. . . . . . . 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 11379	. . . . . 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 462	. . . . 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 3159	. . . 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 7441	. . . . 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 3245	. . . 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 521	. 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 5115	. . . . 5 ⊢ (𝑥 = 𝑅 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))
136	135	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑅 → ((𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅)))
137	136	rexralbidv 3233	. . 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 3584	. 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑅))
140	134, 139	reximddv 3183	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3081 ∃wrex 3091 Vcvv 3457 ⊆ wss 3906 class class class wbr 5111 ⟶wf 6536 ‘cfv 6540 (class class class)co 7416 supcsup 9403 ℝcr 11110 + caddc 11114 · cmul 11116 +∞cpnf 11251 -∞cmnf 11252 ℝ^*cxr 11253 < clt 11254 ≤ cle 11255 − cmin 11452 2c2 12306 3c3 12307 ℝ⁺crp 13027 abscabs 15304 lim supclsp 15540

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 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9405 df-inf 9406 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-n0 12516 df-z 12603 df-uz 12874 df-rp 13028 df-ico 13389 df-seq 14051 df-exp 14111 df-cj 15169 df-re 15170 df-im 15171 df-sqrt 15305 df-abs 15306 df-limsup 15541

This theorem is used by: caurcvgr 15744

W3C validator