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Theorem caubnd 14579
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 14499 . . . 4 ((𝐹𝑘) ∈ ℂ → (abs‘(𝐹𝑘)) ∈ ℝ)
21ralimi 3110 . . 3 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → ∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ)
3 cau3.1 . . . . . . 7 𝑍 = (ℤ𝑀)
43r19.29uz 14571 . . . . . 6 ((∀𝑘𝑍 (𝐹𝑘) ∈ ℂ ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥))
54ex 405 . . . . 5 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)))
65ralimdv 3128 . . . 4 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)))
73caubnd2 14578 . . . 4 (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) → ∃𝑧 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧)
86, 7syl6 35 . . 3 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∃𝑧 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧))
9 fzssuz 12764 . . . . . . . 8 (𝑀...𝑗) ⊆ (ℤ𝑀)
109, 3sseqtr4i 3894 . . . . . . 7 (𝑀...𝑗) ⊆ 𝑍
11 ssralv 3923 . . . . . . 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 13155 . . . . . . . 8 (𝑀...𝑗) ∈ Fin
14 fimaxre3 11388 . . . . . . . 8 (((𝑀...𝑗) ∈ Fin ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥)
1513, 14mpan 677 . . . . . . 7 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥)
16 peano2re 10613 . . . . . . . . . 10 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
1716adantl 474 . . . . . . . . 9 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
18 ltp1 11281 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1))
1918adantl 474 . . . . . . . . . . . . . 14 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 < (𝑥 + 1))
2016adantl 474 . . . . . . . . . . . . . . 15 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
21 lelttr 10531 . . . . . . . . . . . . . . 15 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑥 + 1) ∈ ℝ) → (((abs‘(𝐹𝑘)) ≤ 𝑥𝑥 < (𝑥 + 1)) → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2220, 21mpd3an3 1441 . . . . . . . . . . . . . 14 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘(𝐹𝑘)) ≤ 𝑥𝑥 < (𝑥 + 1)) → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2319, 22mpan2d 681 . . . . . . . . . . . . 13 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2423expcom 406 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → ((abs‘(𝐹𝑘)) ∈ ℝ → ((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1))))
2524ralimdv 3128 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1))))
2625impcom 399 . . . . . . . . . 10 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
27 ralim 3112 . . . . . . . . . 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 brralrspcev 4989 . . . . . . . . 9 (((𝑥 + 1) ∈ ℝ ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < (𝑥 + 1)) → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤)
3017, 28, 29syl6an 671 . . . . . . . 8 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥 → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤))
3130rexlimdva 3229 . . . . . . 7 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → (∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥 → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤))
3215, 31mpd 15 . . . . . 6 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤)
3312, 32syl 17 . . . . 5 (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤)
34 max1 12395 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤))
35343adant3 1112 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤))
36 simp3 1118 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (abs‘(𝐹𝑘)) ∈ ℝ)
37 simp1 1116 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑤 ∈ ℝ)
38 ifcl 4394 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
3938ancoms 451 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
40393adant3 1112 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
41 ltletr 10532 . . . . . . . . . . . . . . . . . 18 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑤𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
4236, 37, 40, 41syl3anc 1351 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑤𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
4335, 42mpan2d 681 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → ((abs‘(𝐹𝑘)) < 𝑤 → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
44 max2 12397 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤))
45443adant3 1112 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤))
46 simp2 1117 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑧 ∈ ℝ)
47 ltletr 10532 . . . . . . . . . . . . . . . . . 18 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑧𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
4836, 46, 40, 47syl3anc 1351 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑧𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
4945, 48mpan2d 681 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → ((abs‘(𝐹𝑘)) < 𝑧 → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5043, 49jaod 845 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
51503expia 1101 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((abs‘(𝐹𝑘)) ∈ ℝ → (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤))))
5251ralimdv 3128 . . . . . . . . . . . . 13 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → ∀𝑘𝑍 (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤))))
53 ralim 3112 . . . . . . . . . . . . 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)))
5452, 53syl6 35 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → ∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤))))
55 brralrspcev 4989 . . . . . . . . . . . . . 14 ((if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ ∧ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)
5655ex 405 . . . . . . . . . . . . 13 (if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ → (∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))
5739, 56syl 17 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))
5854, 57syl6d 75 . . . . . . . . . . 11 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)))
59 uzssz 12078 . . . . . . . . . . . . . . . . . . . . . 22 (ℤ𝑀) ⊆ ℤ
603, 59eqsstri 3891 . . . . . . . . . . . . . . . . . . . . 21 𝑍 ⊆ ℤ
6160sseli 3854 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝑍𝑘 ∈ ℤ)
6260sseli 3854 . . . . . . . . . . . . . . . . . . . 20 (𝑗𝑍𝑗 ∈ ℤ)
63 uztric 12080 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗)))
6461, 62, 63syl2anr 587 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑍𝑘𝑍) → (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗)))
65 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗𝑍𝑘𝑍) → 𝑘𝑍)
6665, 3syl6eleq 2876 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗𝑍𝑘𝑍) → 𝑘 ∈ (ℤ𝑀))
67 elfzuzb 12718 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (𝑀...𝑗) ↔ (𝑘 ∈ (ℤ𝑀) ∧ 𝑗 ∈ (ℤ𝑘)))
6867baib 528 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (ℤ𝑀) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ𝑘)))
6966, 68syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑗𝑍𝑘𝑍) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ𝑘)))
7069orbi1d 900 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑍𝑘𝑍) → ((𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗)) ↔ (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗))))
7164, 70mpbird 249 . . . . . . . . . . . . . . . . . 18 ((𝑗𝑍𝑘𝑍) → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗)))
7271ex 405 . . . . . . . . . . . . . . . . 17 (𝑗𝑍 → (𝑘𝑍 → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗))))
73 pm3.48 946 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)) → ((𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗)) → ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧)))
7472, 73syl9 77 . . . . . . . . . . . . . . . 16 (𝑗𝑍 → (((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)) → (𝑘𝑍 → ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧))))
7574alimdv 1875 . . . . . . . . . . . . . . 15 (𝑗𝑍 → (∀𝑘((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)) → ∀𝑘(𝑘𝑍 → ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧))))
76 df-ral 3093 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ↔ ∀𝑘(𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤))
77 df-ral 3093 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 ↔ ∀𝑘(𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧))
7876, 77anbi12i 617 . . . . . . . . . . . . . . . 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)))
79 19.26 1833 . . . . . . . . . . . . . . . 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)))
8078, 79bitr4i 270 . . . . . . . . . . . . . . 15 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧) ↔ ∀𝑘((𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤) ∧ (𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧)))
81 df-ral 3093 . . . . . . . . . . . . . . 15 (∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) ↔ ∀𝑘(𝑘𝑍 → ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧)))
8275, 80, 813imtr4g 288 . . . . . . . . . . . . . 14 (𝑗𝑍 → ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧) → ∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧)))
83823impib 1096 . . . . . . . . . . . . 13 ((𝑗𝑍 ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧) → ∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧))
8483imim1i 63 . . . . . . . . . . . 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))
85843expd 1333 . . . . . . . . . . 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))))
8658, 85syl6 35 . . . . . . . . . 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)))))
8786com23 86 . . . . . . . . 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)))))
8887expimpd 446 . . . . . . . 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)))))
8988com3r 87 . . . . . . 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)))))
9089com34 91 . . . . . 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)))))
9190rexlimdv 3228 . . . . 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))))
9233, 91mpd 15 . . . 4 (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → ((𝑧 ∈ ℝ ∧ 𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)))
9392rexlimdvv 3238 . . 3 (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → (∃𝑧 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))
942, 8, 93sylsyld 61 . 2 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦))
9594imp 398 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 198  wa 387  wo 833  w3a 1068  wal 1505   = wceq 1507  wcel 2050  wral 3088  wrex 3089  wss 3829  ifcif 4350   class class class wbr 4929  cfv 6188  (class class class)co 6976  Fincfn 8306  cc 10333  cr 10334  1c1 10336   + caddc 10338   < clt 10474  cle 10475  cmin 10670  cz 11793  cuz 12058  +crp 12204  ...cfz 12708  abscabs 14454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412  ax-pre-sup 10413
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-sup 8701  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-div 11099  df-nn 11440  df-2 11503  df-3 11504  df-n0 11708  df-z 11794  df-uz 12059  df-rp 12205  df-fz 12709  df-seq 13185  df-exp 13245  df-cj 14319  df-re 14320  df-im 14321  df-sqrt 14455  df-abs 14456
This theorem is referenced by:  climbdd  14889
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