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Theorem caubnd 15400

Description: A Cauchy sequence of complex numbers is bounded. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 14-Feb-2014.)

**Hypothesis**
Ref	Expression
cau3.1	⊢ 𝑍 = (ℤ_≥‘𝑀)

**Assertion**
Ref	Expression
caubnd	⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (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 caubnd Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		abscl 15319	. . . 4 ⊢ ((𝐹‘𝑘) ∈ ℂ → (abs‘(𝐹‘𝑘)) ∈ ℝ)
2	1	ralimi 3102	. . 3 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ)
3		cau3.1	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4	3	r19.29uz 15392	. . . . . 6 ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
5	4	ex 417	. . . . 5 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
6	5	ralimdv 3179	. . . 4 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
7	3	caubnd2 15399	. . . 4 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑧 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧)
8	6, 7	syl6 36	. . 3 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → ∃𝑧 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧))
9		fzssuz 13584	. . . . . . . 8 ⊢ (𝑀...𝑗) ⊆ (ℤ_≥‘𝑀)
10	9, 3	sseqtrri 3988	. . . . . . 7 ⊢ (𝑀...𝑗) ⊆ 𝑍
11		ssralv 4008	. . . . . . 7 ⊢ ((𝑀...𝑗) ⊆ 𝑍 → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ)
13		fzfi 13999	. . . . . . . 8 ⊢ (𝑀...𝑗) ∈ Fin
14		fimaxre3 12152	. . . . . . . 8 ⊢ (((𝑀...𝑗) ∈ Fin ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ≤ 𝑥)
15	13, 14	mpan 702	. . . . . . 7 ⊢ (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ≤ 𝑥)
16		peano2re 11371	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
17	16	adantl 486	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
18		ltp1 12046	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1))
19	18	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 < (𝑥 + 1))
20	16	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
21		lelttr 11288	. . . . . . . . . . . . . . 15 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑥 + 1) ∈ ℝ) → (((abs‘(𝐹‘𝑘)) ≤ 𝑥 ∧ 𝑥 < (𝑥 + 1)) → (abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
22	20, 21	mpd3an3 1486	. . . . . . . . . . . . . 14 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘(𝐹‘𝑘)) ≤ 𝑥 ∧ 𝑥 < (𝑥 + 1)) → (abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
23	19, 22	mpan2d 706	. . . . . . . . . . . . 13 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((abs‘(𝐹‘𝑘)) ≤ 𝑥 → (abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
24	23	expcom 418	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → ((abs‘(𝐹‘𝑘)) ∈ ℝ → ((abs‘(𝐹‘𝑘)) ≤ 𝑥 → (abs‘(𝐹‘𝑘)) < (𝑥 + 1))))
25	24	ralimdv 3179	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹‘𝑘)) ≤ 𝑥 → (abs‘(𝐹‘𝑘)) < (𝑥 + 1))))
26	25	impcom 412	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹‘𝑘)) ≤ 𝑥 → (abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
27		ralim 3105	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹‘𝑘)) ≤ 𝑥 → (abs‘(𝐹‘𝑘)) < (𝑥 + 1)) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ≤ 𝑥 → ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
28	26, 27	syl 18	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ≤ 𝑥 → ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
29		brralrspcev 5165	. . . . . . . . 9 ⊢ (((𝑥 + 1) ∈ ℝ ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < (𝑥 + 1)) → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤)
30	17, 28, 29	syl6an 696	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ≤ 𝑥 → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤))
31	30	rexlimdva 3166	. . . . . . 7 ⊢ (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ → (∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ≤ 𝑥 → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤))
32	15, 31	mpd 16	. . . . . 6 ⊢ (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤)
33	12, 32	syl 18	. . . . 5 ⊢ (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤)
34		max1 13202	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑤 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
35	34	3adant3 1148	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑤 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
36		simp3 1154	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
37		simp1 1152	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑤 ∈ ℝ)
38		ifcl 4529	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) → if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ)
39	38	ancoms 463	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ)
40	39	3adant3 1148	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ)
41		ltletr 11290	. . . . . . . . . . . . . . . . . 18 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ) → (((abs‘(𝐹‘𝑘)) < 𝑤 ∧ 𝑤 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤)) → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)))
42	36, 37, 40, 41	syl3anc 1394	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → (((abs‘(𝐹‘𝑘)) < 𝑤 ∧ 𝑤 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤)) → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)))
43	35, 42	mpan2d 706	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → ((abs‘(𝐹‘𝑘)) < 𝑤 → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)))
44		max2 13204	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
45	44	3adant3 1148	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑧 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
46		simp2 1153	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑧 ∈ ℝ)
47		ltletr 11290	. . . . . . . . . . . . . . . . . 18 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ) → (((abs‘(𝐹‘𝑘)) < 𝑧 ∧ 𝑧 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤)) → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)))
48	36, 46, 40, 47	syl3anc 1394	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → (((abs‘(𝐹‘𝑘)) < 𝑧 ∧ 𝑧 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤)) → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)))
49	45, 48	mpan2d 706	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → ((abs‘(𝐹‘𝑘)) < 𝑧 → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)))
50	43, 49	jaod 872	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → (((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧) → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)))
51	50	3expia 1137	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((abs‘(𝐹‘𝑘)) ∈ ℝ → (((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧) → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤))))
52	51	ralimdv 3179	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → ∀𝑘 ∈ 𝑍 (((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧) → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤))))
53		ralim 3105	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ 𝑍 (((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧) → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)) → (∀𝑘 ∈ 𝑍 ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧) → ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)))
54	52, 53	syl6 36	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → (∀𝑘 ∈ 𝑍 ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧) → ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤))))
55		brralrspcev 5165	. . . . . . . . . . . . . 14 ⊢ ((if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ ∧ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦)
56	55	ex 417	. . . . . . . . . . . . 13 ⊢ (if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦))
57	39, 56	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦))
58	54, 57	syl6d 76	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → (∀𝑘 ∈ 𝑍 ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦)))
59		uzssz 12874	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
60	3, 59	eqsstri 3985	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 ⊆ ℤ
61	60	sseli 3935	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
62	60	sseli 3935	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
63		uztric 12877	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)))
64	61, 62, 63	syl2anr 608	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑗 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)))
65		simpr 489	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
66	65, 3	eleqtrdi 2875	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (ℤ_≥‘𝑀))
67		elfzuzb 13537	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (𝑀...𝑗) ↔ (𝑘 ∈ (ℤ_≥‘𝑀) ∧ 𝑗 ∈ (ℤ_≥‘𝑘)))
68	67	baib 544	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ_≥‘𝑘)))
69	66, 68	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ_≥‘𝑘)))
70	69	orbi1d 929	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)) ↔ (𝑗 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑗))))
71	64, 70	mpbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)))
72	71	ex 417	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ 𝑍 → (𝑘 ∈ 𝑍 → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ_≥‘𝑗))))
73		pm3.48 978	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹‘𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) → (abs‘(𝐹‘𝑘)) < 𝑧)) → ((𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧)))
74	72, 73	syl9 78	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ 𝑍 → (((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹‘𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) → (abs‘(𝐹‘𝑘)) < 𝑧)) → (𝑘 ∈ 𝑍 → ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧))))
75	74	alimdv 1939	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (∀𝑘((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹‘𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) → (abs‘(𝐹‘𝑘)) < 𝑧)) → ∀𝑘(𝑘 ∈ 𝑍 → ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧))))
76		df-ral 3080	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 ↔ ∀𝑘(𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹‘𝑘)) < 𝑤))
77		df-ral 3080	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 ↔ ∀𝑘(𝑘 ∈ (ℤ_≥‘𝑗) → (abs‘(𝐹‘𝑘)) < 𝑧))
78	76, 77	anbi12i 639	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧) ↔ (∀𝑘(𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹‘𝑘)) < 𝑤) ∧ ∀𝑘(𝑘 ∈ (ℤ_≥‘𝑗) → (abs‘(𝐹‘𝑘)) < 𝑧)))
79		19.26 1893	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹‘𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) → (abs‘(𝐹‘𝑘)) < 𝑧)) ↔ (∀𝑘(𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹‘𝑘)) < 𝑤) ∧ ∀𝑘(𝑘 ∈ (ℤ_≥‘𝑗) → (abs‘(𝐹‘𝑘)) < 𝑧)))
80	78, 79	bitr4i 281	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧) ↔ ∀𝑘((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹‘𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) → (abs‘(𝐹‘𝑘)) < 𝑧)))
81		df-ral 3080	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ 𝑍 ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧) ↔ ∀𝑘(𝑘 ∈ 𝑍 → ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧)))
82	75, 80, 81	3imtr4g 299	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ 𝑍 → ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧) → ∀𝑘 ∈ 𝑍 ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧)))
83	82	3impib 1132	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧) → ∀𝑘 ∈ 𝑍 ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧))
84	83	imim1i 64	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ 𝑍 ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦) → ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦))
85	84	3expd 1370	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ 𝑍 ((abs‘(𝐹‘𝑘)) < 𝑤 ∨ (abs‘(𝐹‘𝑘)) < 𝑧) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦) → (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦))))
86	58, 85	syl6 36	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦)))))
87	86	com23 87	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑗 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦)))))
88	87	expimpd 458	. . . . . . . 8 ⊢ (𝑤 ∈ ℝ → ((𝑧 ∈ ℝ ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦)))))
89	88	com3r 88	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → (𝑤 ∈ ℝ → ((𝑧 ∈ ℝ ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦)))))
90	89	com34 92	. . . . . 6 ⊢ (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → (𝑤 ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 → ((𝑧 ∈ ℝ ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦)))))
91	90	rexlimdv 3164	. . . . 5 ⊢ (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → (∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 → ((𝑧 ∈ ℝ ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦))))
92	33, 91	mpd 16	. . . 4 ⊢ (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → ((𝑧 ∈ ℝ ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦)))
93	92	rexlimdvv 3221	. . 3 ⊢ (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → (∃𝑧 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦))
94	2, 8, 93	sylsyld 62	. 2 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦))
95	94	imp 411	1 ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) < 𝑦)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 ∀wal 1561 = wceq 1563 ∈ wcel 2145 ∀wral 3079 ∃wrex 3089 ⊆ wss 3907 ifcif 4483 class class class wbr 5105 ‘cfv 6525 (class class class)co 7400 Fincfn 8931 ℂcc 11086 ℝcr 11087 1c1 11089 + caddc 11091 < clt 11231 ≤ cle 11232 − cmin 11429 ℤcz 12582 ℤ_≥cuz 12853 ℝ⁺crp 13007 ...cfz 13526 abscabs 15275

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-sup 9390 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-n0 12496 df-z 12583 df-uz 12854 df-rp 13008 df-fz 13527 df-seq 14029 df-exp 14089 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277

This theorem is referenced by: climbdd 15713

W3C validator