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Theorem caubnd 13892
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 = (ℤ𝑀)
Assertion
Ref Expression
caubnd ((∀𝑘𝑍 (𝐹𝑘) ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 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.
StepHypRef Expression
1 abscl 13812 . . . 4 ((𝐹𝑘) ∈ ℂ → (abs‘(𝐹𝑘)) ∈ ℝ)
21ralimi 2935 . . 3 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → ∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ)
3 cau3.1 . . . . . . 7 𝑍 = (ℤ𝑀)
43r19.29uz 13884 . . . . . 6 ((∀𝑘𝑍 (𝐹𝑘) ∈ ℂ ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥))
54ex 448 . . . . 5 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)))
65ralimdv 2945 . . . 4 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)))
73caubnd2 13891 . . . 4 (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) → ∃𝑧 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧)
86, 7syl6 34 . . 3 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∃𝑧 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧))
9 fzssuz 12208 . . . . . . . 8 (𝑀...𝑗) ⊆ (ℤ𝑀)
109, 3sseqtr4i 3600 . . . . . . 7 (𝑀...𝑗) ⊆ 𝑍
11 ssralv 3628 . . . . . . 7 ((𝑀...𝑗) ⊆ 𝑍 → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ))
1210, 11ax-mp 5 . . . . . 6 (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ)
13 fzfi 12588 . . . . . . . 8 (𝑀...𝑗) ∈ Fin
14 fimaxre3 10819 . . . . . . . 8 (((𝑀...𝑗) ∈ Fin ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥)
1513, 14mpan 701 . . . . . . 7 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥)
16 peano2re 10060 . . . . . . . . . 10 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
1716adantl 480 . . . . . . . . 9 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
18 ltp1 10710 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1))
1918adantl 480 . . . . . . . . . . . . . 14 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 < (𝑥 + 1))
2016adantl 480 . . . . . . . . . . . . . . 15 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
21 lelttr 9979 . . . . . . . . . . . . . . 15 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑥 + 1) ∈ ℝ) → (((abs‘(𝐹𝑘)) ≤ 𝑥𝑥 < (𝑥 + 1)) → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2220, 21mpd3an3 1416 . . . . . . . . . . . . . 14 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘(𝐹𝑘)) ≤ 𝑥𝑥 < (𝑥 + 1)) → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2319, 22mpan2d 705 . . . . . . . . . . . . 13 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2423expcom 449 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → ((abs‘(𝐹𝑘)) ∈ ℝ → ((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1))))
2524ralimdv 2945 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1))))
2625impcom 444 . . . . . . . . . 10 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
27 ralim 2931 . . . . . . . . . 10 (∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1)) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥 → ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < (𝑥 + 1)))
2826, 27syl 17 . . . . . . . . 9 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥 → ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < (𝑥 + 1)))
29 breq2 4581 . . . . . . . . . . 11 (𝑤 = (𝑥 + 1) → ((abs‘(𝐹𝑘)) < 𝑤 ↔ (abs‘(𝐹𝑘)) < (𝑥 + 1)))
3029ralbidv 2968 . . . . . . . . . 10 (𝑤 = (𝑥 + 1) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ↔ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < (𝑥 + 1)))
3130rspcev 3281 . . . . . . . . 9 (((𝑥 + 1) ∈ ℝ ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < (𝑥 + 1)) → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤)
3217, 28, 31syl6an 565 . . . . . . . 8 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥 → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤))
3332rexlimdva 3012 . . . . . . 7 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → (∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥 → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤))
3415, 33mpd 15 . . . . . 6 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤)
3512, 34syl 17 . . . . 5 (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤)
36 max1 11849 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤))
37363adant3 1073 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤))
38 simp3 1055 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (abs‘(𝐹𝑘)) ∈ ℝ)
39 simp1 1053 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑤 ∈ ℝ)
40 ifcl 4079 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
4140ancoms 467 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
42413adant3 1073 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
43 ltletr 9980 . . . . . . . . . . . . . . . . . 18 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑤𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
4438, 39, 42, 43syl3anc 1317 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑤𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
4537, 44mpan2d 705 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → ((abs‘(𝐹𝑘)) < 𝑤 → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
46 max2 11851 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤))
47463adant3 1073 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤))
48 simp2 1054 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑧 ∈ ℝ)
49 ltletr 9980 . . . . . . . . . . . . . . . . . 18 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑧𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5038, 48, 42, 49syl3anc 1317 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑧𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5147, 50mpan2d 705 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → ((abs‘(𝐹𝑘)) < 𝑧 → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5245, 51jaod 393 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
53523expia 1258 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((abs‘(𝐹𝑘)) ∈ ℝ → (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤))))
5453ralimdv 2945 . . . . . . . . . . . . 13 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → ∀𝑘𝑍 (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤))))
55 ralim 2931 . . . . . . . . . . . . 13 (∀𝑘𝑍 (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)) → (∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → ∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5654, 55syl6 34 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → ∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤))))
57 breq2 4581 . . . . . . . . . . . . . . . 16 (𝑦 = if(𝑤𝑧, 𝑧, 𝑤) → ((abs‘(𝐹𝑘)) < 𝑦 ↔ (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5857ralbidv 2968 . . . . . . . . . . . . . . 15 (𝑦 = if(𝑤𝑧, 𝑧, 𝑤) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦 ↔ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5958rspcev 3281 . . . . . . . . . . . . . 14 ((if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ ∧ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)
6059ex 448 . . . . . . . . . . . . 13 (if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ → (∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))
6141, 60syl 17 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))
6256, 61syl6d 72 . . . . . . . . . . 11 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)))
63 uzssz 11539 . . . . . . . . . . . . . . . . . . . . . 22 (ℤ𝑀) ⊆ ℤ
643, 63eqsstri 3597 . . . . . . . . . . . . . . . . . . . . 21 𝑍 ⊆ ℤ
6564sseli 3563 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝑍𝑘 ∈ ℤ)
6664sseli 3563 . . . . . . . . . . . . . . . . . . . 20 (𝑗𝑍𝑗 ∈ ℤ)
67 uztric 11541 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗)))
6865, 66, 67syl2anr 493 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑍𝑘𝑍) → (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗)))
69 simpr 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗𝑍𝑘𝑍) → 𝑘𝑍)
7069, 3syl6eleq 2697 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗𝑍𝑘𝑍) → 𝑘 ∈ (ℤ𝑀))
71 elfzuzb 12162 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (𝑀...𝑗) ↔ (𝑘 ∈ (ℤ𝑀) ∧ 𝑗 ∈ (ℤ𝑘)))
7271baib 941 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (ℤ𝑀) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ𝑘)))
7370, 72syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑗𝑍𝑘𝑍) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ𝑘)))
7473orbi1d 734 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑍𝑘𝑍) → ((𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗)) ↔ (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗))))
7568, 74mpbird 245 . . . . . . . . . . . . . . . . . 18 ((𝑗𝑍𝑘𝑍) → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗)))
7675ex 448 . . . . . . . . . . . . . . . . 17 (𝑗𝑍 → (𝑘𝑍 → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗))))
77 pm3.48 873 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)) → ((𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗)) → ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧)))
7876, 77syl9 74 . . . . . . . . . . . . . . . 16 (𝑗𝑍 → (((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)) → (𝑘𝑍 → ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧))))
7978alimdv 1831 . . . . . . . . . . . . . . 15 (𝑗𝑍 → (∀𝑘((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)) → ∀𝑘(𝑘𝑍 → ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧))))
80 df-ral 2900 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ↔ ∀𝑘(𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤))
81 df-ral 2900 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 ↔ ∀𝑘(𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧))
8280, 81anbi12i 728 . . . . . . . . . . . . . . . 16 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧) ↔ (∀𝑘(𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ ∀𝑘(𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)))
83 19.26 1785 . . . . . . . . . . . . . . . 16 (∀𝑘((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)) ↔ (∀𝑘(𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ ∀𝑘(𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)))
8482, 83bitr4i 265 . . . . . . . . . . . . . . 15 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧) ↔ ∀𝑘((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)))
85 df-ral 2900 . . . . . . . . . . . . . . 15 (∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) ↔ ∀𝑘(𝑘𝑍 → ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧)))
8679, 84, 853imtr4g 283 . . . . . . . . . . . . . 14 (𝑗𝑍 → ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧) → ∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧)))
87863impib 1253 . . . . . . . . . . . . 13 ((𝑗𝑍 ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧) → ∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧))
8887imim1i 60 . . . . . . . . . . . 12 ((∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦) → ((𝑗𝑍 ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))
89883expd 1275 . . . . . . . . . . 11 ((∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦) → (𝑗𝑍 → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 → (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))))
9062, 89syl6 34 . . . . . . . . . 10 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (𝑗𝑍 → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 → (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)))))
9190com23 83 . . . . . . . . 9 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑗𝑍 → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 → (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)))))
9291expimpd 626 . . . . . . . 8 (𝑤 ∈ ℝ → ((𝑧 ∈ ℝ ∧ 𝑗𝑍) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 → (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)))))
9392com3r 84 . . . . . . 7 (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (𝑤 ∈ ℝ → ((𝑧 ∈ ℝ ∧ 𝑗𝑍) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 → (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)))))
9493com34 88 . . . . . 6 (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (𝑤 ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 → ((𝑧 ∈ ℝ ∧ 𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)))))
9594rexlimdv 3011 . . . . 5 (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 → ((𝑧 ∈ ℝ ∧ 𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))))
9635, 95mpd 15 . . . 4 (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → ((𝑧 ∈ ℝ ∧ 𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)))
9796rexlimdvv 3018 . . 3 (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (∃𝑧 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))
982, 8, 97sylsyld 58 . 2 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))
9998imp 443 1 ((∀𝑘𝑍 (𝐹𝑘) ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382  w3a 1030  wal 1472   = wceq 1474  wcel 1976  wral 2895  wrex 2896  wss 3539  ifcif 4035   class class class wbr 4577  cfv 5790  (class class class)co 6527  Fincfn 7818  cc 9790  cr 9791  1c1 9793   + caddc 9795   < clt 9930  cle 9931  cmin 10117  cz 11210  cuz 11519  +crp 11664  ...cfz 12152  abscabs 13768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-rp 11665  df-fz 12153  df-seq 12619  df-exp 12678  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770
This theorem is referenced by:  climbdd  14196
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