Theorem caubnd2 14097

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 11836	. . 3 ⊢ 1 ∈ ℝ+
2		breq2 4657	. . . . . 6 ⊢ (𝑥 = 1 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1))
3	2	anbi2d 740	. . . . 5 ⊢ (𝑥 = 1 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)))
4	3	rexralbidv 3058	. . . 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 3305	. . 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 11697	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ)
8		cau3.1	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ≥‘𝑀)
9	7, 8	eleq2s 2719	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
10		uzid 11702	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ≥‘𝑗))
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑗))
12		simpl 473	. . . . . . . . . 10 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → (𝐹‘𝑘) ∈ ℂ)
13	12	ralimi 2952	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ)
14		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
15	14	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ))
16	15	rspcva 3307	. . . . . . . . 9 ⊢ ((𝑗 ∈ (ℤ≥‘𝑗) ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ ℂ) → (𝐹‘𝑗) ∈ ℂ)
17	11, 13, 16	syl2an 494	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)) → (𝐹‘𝑗) ∈ ℂ)
18		abscl 14018	. . . . . . . 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 10039	. . . . . . 7 ⊢ 1 ∈ ℝ
21		readdcl 10019	. . . . . . 7 ⊢ (((abs‘(𝐹‘𝑗)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(𝐹‘𝑗)) + 1) ∈ ℝ)
22	19, 20, 21	sylancl 694	. . . . . 6 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)) → ((abs‘(𝐹‘𝑗)) + 1) ∈ ℝ)
23		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → (𝐹‘𝑘) ∈ ℂ)
24		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → (𝐹‘𝑗) ∈ ℂ)
25		abs2dif 14072	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → ((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))))
26	23, 24, 25	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → ((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))))
27		abscl 14018	. . . . . . . . . . . . . . 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 10458	. . . . . . . . . . . . 13 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → ((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) ∈ ℝ)
31	23, 24	subcld 10392	. . . . . . . . . . . . . 14 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → ((𝐹‘𝑘) − (𝐹‘𝑗)) ∈ ℂ)
32		abscl 14018	. . . . . . . . . . . . . 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 10128	. . . . . . . . . . . . . 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 1413	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 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 711	. . . . . . . . . . 11 ⊢ (((𝑗 ∈ 𝑍 ∧ (𝐹‘𝑗) ∈ ℂ) ∧ (𝐹‘𝑘) ∈ ℂ) → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1 → ((abs‘(𝐹‘𝑘)) − (abs‘(𝐹‘𝑗))) < 1))
38		ltsubadd2 10499	. . . . . . . . . . . . 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 1413	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 229	. . . . . . . . . 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 629	. . . . . . . . 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 2963	. . . . . . . 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 456	. . . . . . 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		breq2 4657	. . . . . . . 8 ⊢ (𝑦 = ((abs‘(𝐹‘𝑗)) + 1) → ((abs‘(𝐹‘𝑘)) < 𝑦 ↔ (abs‘(𝐹‘𝑘)) < ((abs‘(𝐹‘𝑗)) + 1)))
47	46	ralbidv 2986	. . . . . . 7 ⊢ (𝑦 = ((abs‘(𝐹‘𝑗)) + 1) → (∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < ((abs‘(𝐹‘𝑗)) + 1)))
48	47	rspcev 3309	. . . . . 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)
49	22, 45, 48	syl2anc 693	. . . . 5 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦)
50	49	ex 450	. . . 4 ⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 1) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦))
51	50	reximia 3009	. . 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)
52		rexcom 3099	. . 3 ⊢ (∃𝑗 ∈ 𝑍 ∃𝑦 ∈ ℝ ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦 ↔ ∃𝑦 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑦)
53	51, 52	sylib 208	. 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)
54	6, 53	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  ℝcr 9935  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266  ℤcz 11377  ℤ_≥cuz 11687  ℝ⁺crp 11832  abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by:  caubnd  14098