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Theorem caubnd 13946
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 13866 . . . 4 ((𝐹𝑘) ∈ ℂ → (abs‘(𝐹𝑘)) ∈ ℝ)
21ralimi 2936 . . 3 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → ∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ)
3 cau3.1 . . . . . . 7 𝑍 = (ℤ𝑀)
43r19.29uz 13938 . . . . . 6 ((∀𝑘𝑍 (𝐹𝑘) ∈ ℂ ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥))
54ex 449 . . . . 5 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)))
65ralimdv 2946 . . . 4 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)))
73caubnd2 13945 . . . 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 12253 . . . . . . . 8 (𝑀...𝑗) ⊆ (ℤ𝑀)
109, 3sseqtr4i 3601 . . . . . . 7 (𝑀...𝑗) ⊆ 𝑍
11 ssralv 3629 . . . . . . 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 12633 . . . . . . . 8 (𝑀...𝑗) ∈ Fin
14 fimaxre3 10849 . . . . . . . 8 (((𝑀...𝑗) ∈ Fin ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥)
1513, 14mpan 702 . . . . . . 7 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥)
16 peano2re 10088 . . . . . . . . . 10 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
1716adantl 481 . . . . . . . . 9 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
18 ltp1 10740 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1))
1918adantl 481 . . . . . . . . . . . . . 14 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 < (𝑥 + 1))
2016adantl 481 . . . . . . . . . . . . . . 15 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
21 lelttr 10007 . . . . . . . . . . . . . . 15 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑥 + 1) ∈ ℝ) → (((abs‘(𝐹𝑘)) ≤ 𝑥𝑥 < (𝑥 + 1)) → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2220, 21mpd3an3 1417 . . . . . . . . . . . . . 14 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘(𝐹𝑘)) ≤ 𝑥𝑥 < (𝑥 + 1)) → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2319, 22mpan2d 706 . . . . . . . . . . . . 13 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2423expcom 450 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → ((abs‘(𝐹𝑘)) ∈ ℝ → ((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1))))
2524ralimdv 2946 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1))))
2625impcom 445 . . . . . . . . . 10 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
27 ralim 2932 . . . . . . . . . 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 4587 . . . . . . . . . . 11 (𝑤 = (𝑥 + 1) → ((abs‘(𝐹𝑘)) < 𝑤 ↔ (abs‘(𝐹𝑘)) < (𝑥 + 1)))
3029ralbidv 2969 . . . . . . . . . 10 (𝑤 = (𝑥 + 1) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ↔ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < (𝑥 + 1)))
3130rspcev 3282 . . . . . . . . 9 (((𝑥 + 1) ∈ ℝ ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < (𝑥 + 1)) → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤)
3217, 28, 31syl6an 566 . . . . . . . 8 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥 → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤))
3332rexlimdva 3013 . . . . . . 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 11890 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤))
37363adant3 1074 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤))
38 simp3 1056 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (abs‘(𝐹𝑘)) ∈ ℝ)
39 simp1 1054 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑤 ∈ ℝ)
40 ifcl 4080 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
4140ancoms 468 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
42413adant3 1074 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
43 ltletr 10008 . . . . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑤𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
4537, 44mpan2d 706 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → ((abs‘(𝐹𝑘)) < 𝑤 → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
46 max2 11892 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤))
47463adant3 1074 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤))
48 simp2 1055 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑧 ∈ ℝ)
49 ltletr 10008 . . . . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑧𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5147, 50mpan2d 706 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → ((abs‘(𝐹𝑘)) < 𝑧 → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5245, 51jaod 394 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
53523expia 1259 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((abs‘(𝐹𝑘)) ∈ ℝ → (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤))))
5453ralimdv 2946 . . . . . . . . . . . . 13 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → ∀𝑘𝑍 (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤))))
55 ralim 2932 . . . . . . . . . . . . 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 4587 . . . . . . . . . . . . . . . 16 (𝑦 = if(𝑤𝑧, 𝑧, 𝑤) → ((abs‘(𝐹𝑘)) < 𝑦 ↔ (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5857ralbidv 2969 . . . . . . . . . . . . . . 15 (𝑦 = if(𝑤𝑧, 𝑧, 𝑤) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦 ↔ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5958rspcev 3282 . . . . . . . . . . . . . 14 ((if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ ∧ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)
6059ex 449 . . . . . . . . . . . . 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 73 . . . . . . . . . . 11 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)))
63 uzssz 11583 . . . . . . . . . . . . . . . . . . . . . 22 (ℤ𝑀) ⊆ ℤ
643, 63eqsstri 3598 . . . . . . . . . . . . . . . . . . . . 21 𝑍 ⊆ ℤ
6564sseli 3564 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝑍𝑘 ∈ ℤ)
6664sseli 3564 . . . . . . . . . . . . . . . . . . . 20 (𝑗𝑍𝑗 ∈ ℤ)
67 uztric 11585 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗)))
6865, 66, 67syl2anr 494 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑍𝑘𝑍) → (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗)))
69 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗𝑍𝑘𝑍) → 𝑘𝑍)
7069, 3syl6eleq 2698 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗𝑍𝑘𝑍) → 𝑘 ∈ (ℤ𝑀))
71 elfzuzb 12207 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (𝑀...𝑗) ↔ (𝑘 ∈ (ℤ𝑀) ∧ 𝑗 ∈ (ℤ𝑘)))
7271baib 942 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (ℤ𝑀) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ𝑘)))
7370, 72syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑗𝑍𝑘𝑍) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ𝑘)))
7473orbi1d 735 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑍𝑘𝑍) → ((𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗)) ↔ (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗))))
7568, 74mpbird 246 . . . . . . . . . . . . . . . . . 18 ((𝑗𝑍𝑘𝑍) → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗)))
7675ex 449 . . . . . . . . . . . . . . . . 17 (𝑗𝑍 → (𝑘𝑍 → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗))))
77 pm3.48 874 . . . . . . . . . . . . . . . . 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 75 . . . . . . . . . . . . . . . 16 (𝑗𝑍 → (((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)) → (𝑘𝑍 → ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧))))
7978alimdv 1832 . . . . . . . . . . . . . . 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 2901 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ↔ ∀𝑘(𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤))
81 df-ral 2901 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 ↔ ∀𝑘(𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧))
8280, 81anbi12i 729 . . . . . . . . . . . . . . . 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 1786 . . . . . . . . . . . . . . . 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 266 . . . . . . . . . . . . . . 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 2901 . . . . . . . . . . . . . . 15 (∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) ↔ ∀𝑘(𝑘𝑍 → ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧)))
8679, 84, 853imtr4g 284 . . . . . . . . . . . . . 14 (𝑗𝑍 → ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧) → ∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧)))
87863impib 1254 . . . . . . . . . . . . 13 ((𝑗𝑍 ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧) → ∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧))
8887imim1i 61 . . . . . . . . . . . 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 1276 . . . . . . . . . . 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 84 . . . . . . . . 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 627 . . . . . . . 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 85 . . . . . . 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 89 . . . . . 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 3012 . . . . 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 3019 . . 3 (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (∃𝑧 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))
982, 8, 97sylsyld 59 . 2 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))
9998imp 444 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 195  wo 382  wa 383  w3a 1031  wal 1473   = wceq 1475  wcel 1977  wral 2896  wrex 2897  wss 3540  ifcif 4036   class class class wbr 4583  cfv 5804  (class class class)co 6549  Fincfn 7841  cc 9813  cr 9814  1c1 9816   + caddc 9818   < clt 9953  cle 9954  cmin 10145  cz 11254  cuz 11563  +crp 11708  ...cfz 12197  abscabs 13822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824
This theorem is referenced by:  climbdd  14250
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