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Theorem caubnd 14998

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 14918	. . . 4 ⊢ ((𝐹‘𝑘) ∈ ℂ → (abs‘(𝐹‘𝑘)) ∈ ℝ)
2	1	ralimi 3086	. . 3 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ∈ ℝ)
3		cau3.1	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4	3	r19.29uz 14990	. . . . . 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 412	. . . . 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 3103	. . . 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 14997	. . . 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 13226	. . . . . . . 8 ⊢ (𝑀...𝑗) ⊆ (ℤ_≥‘𝑀)
10	9, 3	sseqtrri 3954	. . . . . . 7 ⊢ (𝑀...𝑗) ⊆ 𝑍
11		ssralv 3983	. . . . . . 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 13620	. . . . . . . 8 ⊢ (𝑀...𝑗) ∈ Fin
14		fimaxre3 11851	. . . . . . . 8 ⊢ (((𝑀...𝑗) ∈ Fin ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ≤ 𝑥)
15	13, 14	mpan 686	. . . . . . 7 ⊢ (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ≤ 𝑥)
16		peano2re 11078	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
17	16	adantl 481	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
18		ltp1 11745	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1))
19	18	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 < (𝑥 + 1))
20	16	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
21		lelttr 10996	. . . . . . . . . . . . . . 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 1460	. . . . . . . . . . . . . 14 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘(𝐹‘𝑘)) ≤ 𝑥 ∧ 𝑥 < (𝑥 + 1)) → (abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
23	19, 22	mpan2d 690	. . . . . . . . . . . . 13 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((abs‘(𝐹‘𝑘)) ≤ 𝑥 → (abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
24	23	expcom 413	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → ((abs‘(𝐹‘𝑘)) ∈ ℝ → ((abs‘(𝐹‘𝑘)) ≤ 𝑥 → (abs‘(𝐹‘𝑘)) < (𝑥 + 1))))
25	24	ralimdv 3103	. . . . . . . . . . 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 407	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹‘𝑘)) ≤ 𝑥 → (abs‘(𝐹‘𝑘)) < (𝑥 + 1)))
27		ralim 3082	. . . . . . . . . 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 5130	. . . . . . . . 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 680	. . . . . . . 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 3212	. . . . . . 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 12848	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑤 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
35	34	3adant3 1130	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑤 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
36		simp3 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
37		simp1 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑤 ∈ ℝ)
38		ifcl 4501	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) → if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ)
39	38	ancoms 458	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ)
40	39	3adant3 1130	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → if(𝑤 ≤ 𝑧, 𝑧, 𝑤) ∈ ℝ)
41		ltletr 10997	. . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → ((abs‘(𝐹‘𝑘)) < 𝑤 → (abs‘(𝐹‘𝑘)) < if(𝑤 ≤ 𝑧, 𝑧, 𝑤)))
44		max2 12850	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
45	44	3adant3 1130	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑧 ≤ if(𝑤 ≤ 𝑧, 𝑧, 𝑤))
46		simp2 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹‘𝑘)) ∈ ℝ) → 𝑧 ∈ ℝ)
47		ltletr 10997	. . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . . 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 3103	. . . . . . . . . . . . 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 3082	. . . . . . . . . . . . 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 5130	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 12532	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
60	3, 59	eqsstri 3951	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 ⊆ ℤ
61	60	sseli 3913	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
62	60	sseli 3913	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
63		uztric 12535	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)))
64	61, 62, 63	syl2anr 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑗 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)))
65		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
66	65, 3	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (ℤ_≥‘𝑀))
67		elfzuzb 13179	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (𝑀...𝑗) ↔ (𝑘 ∈ (ℤ_≥‘𝑀) ∧ 𝑗 ∈ (ℤ_≥‘𝑘)))
68	67	baib 535	. . . . . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ_≥‘𝑗)))
72	71	ex 412	. . . . . . . . . . . . . . . . 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 1920	. . . . . . . . . . . . . . 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 3068	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹‘𝑘)) < 𝑤 ↔ ∀𝑘(𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹‘𝑘)) < 𝑤))
77		df-ral 3068	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(𝐹‘𝑘)) < 𝑧 ↔ ∀𝑘(𝑘 ∈ (ℤ_≥‘𝑗) → (abs‘(𝐹‘𝑘)) < 𝑧))
78	76, 77	anbi12i 626	. . . . . . . . . . . . . . . 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 1874	. . . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . . 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 3068	. . . . . . . . . . . . . . 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 295	. . . . . . . . . . . . . 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 1114	. . . . . . . . . . . . 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 1351	. . . . . . . . . . 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 453	. . . . . . . 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 3211	. . . . 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 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 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 406	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 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 ∀wal 1537 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 ⊆ wss 3883 ifcif 4456 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ℂcc 10800 ℝcr 10801 1c1 10803 + caddc 10805 < clt 10940 ≤ cle 10941 − cmin 11135 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 ...cfz 13168 abscabs 14873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875

This theorem is referenced by: climbdd 15311

W3C validator