Theorem caubnd2 15320

Description: A Cauchy sequence of complex numbers is eventually bounded. (Contributed by Mario Carneiro, 14-Feb-2014.)

Hypothesis

Ref	Expression
cau3.1	⊢ 𝑍 = (ℤ≥‘𝑀)

Assertion

Ref	Expression
caubnd2	⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦)

Distinct variable groups:   𝑗,𝑘,𝑥,𝑦,𝐹   𝑗,𝑀,𝑘,𝑥   𝑗,𝑍,𝑘,𝑥,𝑦

Allowed substitution hint:   𝑀(𝑦)

Proof of Theorem caubnd2

Step	Hyp	Ref	Expression
1		1rp 12946	. . 3 ⊢ 1 ∈ ℝ+
2		breq2 5089	. . . . . 6 ⊢ (𝑥 = 1 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1))
3	2	anbi2d 631	. . . . 5 ⊢ (𝑥 = 1 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)))
4	3	rexralbidv 3203	. . . 4 ⊢ (𝑥 = 1 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)))
5	4	rspcv 3560	. . 3 ⊢ (1 ∈ ℝ+ → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)))
6	1, 5	ax-mp 5	. 2 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1))
7		eluzelz 12798	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ)
8		cau3.1	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ≥‘𝑀)
9	7, 8	eleq2s 2854	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
10		uzid 12803	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ≥‘𝑗))
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑗))
12		simpl 482	. . . . . . . . . 10 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → (𝐹‘𝑘) ∈ ℂ)
13	12	ralimi 3074	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
14		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
15	14	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ))
16	15	rspcva 3562	. . . . . . . . 9 ⊢ ((𝑗 ∈ (ℤ≥‘𝑗) ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ) → (𝐹‘𝑗) ∈ ℂ)
17	11, 13, 16	syl2an 597	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)) → (𝐹‘𝑗) ∈ ℂ)
18		abscl 15240	. . . . . . . 8 ⊢ ((𝐹‘𝑗) ∈ ℂ → (abs‘(𝐹‘𝑗)) ∈ ℝ)
19	17, 18	syl 17	. . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)) → (abs‘(𝐹‘𝑗)) ∈ ℝ)
20		1re 11144	. . . . . . 7 ⊢ 1 ∈ ℝ
21		readdcl 11121	. . . . . . 7 ⊢ (((abs‘(𝐹‘𝑗)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(𝐹‘𝑗)) + 1) ∈ ℝ)
22	19, 20, 21	sylancl 587	. . . . . 6 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)) → ((abs‘(𝐹‘𝑗)) + 1) ∈ ℝ)
23		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → (𝐹‘𝑘) ∈ ℂ)
24		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → (𝐹‘𝑗) ∈ ℂ)
25		abs2dif 15295	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → ((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))))
26	23, 24, 25	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → ((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))))
27		abscl 15240	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) ∈ ℂ → (abs‘(𝐹‘𝑘)) ∈ ℝ)
28	23, 27	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
29	24, 18	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → (abs‘(𝐹‘𝑗)) ∈ ℝ)
30	28, 29	resubcld 11578	. . . . . . . . . . . . 13 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → ((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) ∈ ℝ)
31	23, 24	subcld 11505	. . . . . . . . . . . . . 14 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → ((𝐹‘𝑘) − (𝐹‘𝑗)) ∈ ℂ)
32		abscl 15240	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) − (𝐹‘𝑗)) ∈ ℂ → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∈ ℝ)
33	31, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∈ ℝ)
34		lelttr 11236	. . . . . . . . . . . . . 14 ⊢ ((((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∈ ℝ ∧ 1 ∈ ℝ) → ((((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → ((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) < 1))
35	20, 34	mp3an3 1453	. . . . . . . . . . . . 13 ⊢ ((((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∈ ℝ) → ((((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → ((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) < 1))
36	30, 33, 35	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → ((((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → ((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) < 1))
37	26, 36	mpand 696	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1 → ((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) < 1))
38		ltsubadd2 11621	. . . . . . . . . . . . 13 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ (abs‘(𝐹‘𝑗)) ∈ ℝ ∧ 1 ∈ ℝ) → (((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) < 1 ↔ (abs‘(𝐹‘𝑘)) < ((abs‘(𝐹‘𝑗)) + 1)))
39	20, 38	mp3an3 1453	. . . . . . . . . . . 12 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ (abs‘(𝐹‘𝑗)) ∈ ℝ) → (((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) < 1 ↔ (abs‘(𝐹‘𝑘)) < ((abs‘(𝐹‘𝑗)) + 1)))
40	28, 29, 39	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → (((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) < 1 ↔ (abs‘(𝐹‘𝑘)) < ((abs‘(𝐹‘𝑗)) + 1)))
41	37, 40	sylibd 239	. . . . . . . . . 10 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1 → (abs‘(𝐹‘𝑘)) < ((abs‘(𝐹‘𝑗)) + 1)))
42	41	expimpd 453	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → (abs‘(𝐹‘𝑘)) < ((abs‘(𝐹‘𝑗)) + 1)))
43	42	ralimdv 3151	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < ((abs‘(𝐹‘𝑗)) + 1)))
44	43	impancom 451	. . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)) → ((𝐹‘𝑗) ∈ ℂ → ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < ((abs‘(𝐹‘𝑗)) + 1)))
45	17, 44	mpd 15	. . . . . 6 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)) → ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < ((abs‘(𝐹‘𝑗)) + 1))
46		brralrspcev 5145	. . . . . 6 ⊢ ((((abs‘(𝐹‘𝑗)) + 1) ∈ ℝ ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < ((abs‘(𝐹‘𝑗)) + 1)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦)
47	22, 45, 46	syl2anc 585	. . . . 5 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦)
48	47	ex 412	. . . 4 ⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦))
49	48	reximia 3072	. . 3 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → ∃𝑗 ∈ 𝑍 ∃𝑦 ∈ ℝ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦)
50		rexcom 3266	. . 3 ⊢ (∃𝑗 ∈ 𝑍 ∃𝑦 ∈ ℝ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦 ↔ ∃𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦)
51	49, 50	sylib 218	. 2 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → ∃𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦)
52	6, 51	syl 17	1 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  ℂcc 11036  ℝcr 11037  1c1 11039   + caddc 11041   < clt 11179   ≤ cle 11180   − cmin 11377  ℤcz 12524  ℤ_≥cuz 12788  ℝ⁺crp 12942  abscabs 15196

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-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-seq 13964  df-exp 14024  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198

This theorem is referenced by:  caubnd  15321