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Theorem caubnd 14720

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 14640	. . . 4 ⊢ ((𝐹‘𝑘) ∈ ℂ → (abs‘(𝐹‘𝑘)) ∈ ℝ)
2	1	ralimi 3162	. . 3 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ)
3		cau3.1	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4	3	r19.29uz 14712	. . . . . 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 415	. . . . 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 3180	. . . 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 14719	. . . 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 35	. . 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 12951	. . . . . . . 8 ⊢ (𝑀...𝑗) ⊆ (ℤ_≥‘𝑀)
10	9, 3	sseqtrri 4006	. . . . . . 7 ⊢ (𝑀...𝑗) ⊆ 𝑍
11		ssralv 4035	. . . . . . 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 13343	. . . . . . . 8 ⊢ (𝑀...𝑗) ∈ Fin
14		fimaxre3 11589	. . . . . . . 8 ⊢ (((𝑀...𝑗) ∈ Fin ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ≤ 𝑥)
15	13, 14	mpan 688	. . . . . . 7 ⊢ (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ≤ 𝑥)
16		peano2re 10815	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
17	16	adantl 484	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
18		ltp1 11482	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1))
19	18	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 < (𝑥 + 1))
20	16	adantl 484	. . . . . . . . . . . . . . 15 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
21		lelttr 10733	. . . . . . . . . . . . . . 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 1458	. . . . . . . . . . . . . 14 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘(𝐹‘𝑘)) ≤ 𝑥 ∧ 𝑥 < (𝑥 + 1)) → (abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
23	19, 22	mpan2d 692	. . . . . . . . . . . . 13 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((abs‘(𝐹‘𝑘)) ≤ 𝑥 → (abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
24	23	expcom 416	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → ((abs‘(𝐹‘𝑘)) ∈ ℝ → ((abs‘(𝐹‘𝑘)) ≤ 𝑥 → (abs‘(𝐹‘𝑘)) < (𝑥 + 1))))
25	24	ralimdv 3180	. . . . . . . . . . 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 410	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹‘𝑘)) ≤ 𝑥 → (abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
27		ralim 3164	. . . . . . . . . 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 17	. . . . . . . . 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 5128	. . . . . . . . 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 682	. . . . . . . 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 3286	. . . . . . 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 15	. . . . . 6 ⊢ (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤)
33	12, 32	syl 17	. . . . 5 ⊢ (∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤)
34		max1 12581	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑤 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
35	34	3adant3 1128	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑤 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
36		simp3 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
37		simp1 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑤 ∈ ℝ)
38		ifcl 4513	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) → if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ)
39	38	ancoms 461	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ)
40	39	3adant3 1128	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ)
41		ltletr 10734	. . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → ((abs‘(𝐹‘𝑘)) < 𝑤 → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)))
44		max2 12583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
45	44	3adant3 1128	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑧 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
46		simp2 1133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑧 ∈ ℝ)
47		ltletr 10734	. . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . . 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 3180	. . . . . . . . . . . . 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 3164	. . . . . . . . . . . . 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 35	. . . . . . . . . . . 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 5128	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 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 75	. . . . . . . . . . 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 12267	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
60	3, 59	eqsstri 4003	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 ⊆ ℤ
61	60	sseli 3965	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
62	60	sseli 3965	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
63		uztric 12269	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)))
64	61, 62, 63	syl2anr 598	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑗 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)))
65		simpr 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
66	65, 3	eleqtrdi 2925	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (ℤ_≥‘𝑀))
67		elfzuzb 12905	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (𝑀...𝑗) ↔ (𝑘 ∈ (ℤ_≥‘𝑀) ∧ 𝑗 ∈ (ℤ_≥‘𝑘)))
68	67	baib 538	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ_≥‘𝑘)))
69	66, 68	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ_≥‘𝑘)))
70	69	orbi1d 913	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)) ↔ (𝑗 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑗))))
71	64, 70	mpbird 259	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)))
72	71	ex 415	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ 𝑍 → (𝑘 ∈ 𝑍 → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ_≥‘𝑗))))
73		pm3.48 960	. . . . . . . . . . . . . . . . 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 77	. . . . . . . . . . . . . . . 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 1917	. . . . . . . . . . . . . . 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 3145	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 ↔ ∀𝑘(𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹‘𝑘)) < 𝑤))
77		df-ral 3145	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 ↔ ∀𝑘(𝑘 ∈ (ℤ_≥‘𝑗) → (abs‘(𝐹‘𝑘)) < 𝑧))
78	76, 77	anbi12i 628	. . . . . . . . . . . . . . . 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 1871	. . . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . . 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 3145	. . . . . . . . . . . . . . 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 298	. . . . . . . . . . . . . 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 1112	. . . . . . . . . . . . 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 63	. . . . . . . . . . . 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 1349	. . . . . . . . . . 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 35	. . . . . . . . . 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 86	. . . . . . . . 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 456	. . . . . . . 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 87	. . . . . . 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 91	. . . . . 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 3285	. . . . 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 15	. . . 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 3295	. . 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 61	. 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 409	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 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 ifcif 4469 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℂcc 10537 ℝcr 10538 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 − cmin 10872 ℤcz 11984 ℤ_≥cuz 12246 ℝ⁺crp 12392 ...cfz 12895 abscabs 14595

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597

This theorem is referenced by: climbdd 15030

W3C validator