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Theorem caubnd 15341
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 15261 . . . 4 ((𝐹𝑘) ∈ ℂ → (abs‘(𝐹𝑘)) ∈ ℝ)
21ralimi 3072 . . 3 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → ∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ)
3 cau3.1 . . . . . . 7 𝑍 = (ℤ𝑀)
43r19.29uz 15333 . . . . . 6 ((∀𝑘𝑍 (𝐹𝑘) ∈ ℂ ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥))
54ex 411 . . . . 5 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)))
65ralimdv 3158 . . . 4 (∀𝑘𝑍 (𝐹𝑘) ∈ ℂ → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℂ ∧ (abs‘((𝐹𝑘) − (𝐹𝑗))) < 𝑥)))
73caubnd2 15340 . . . 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 13577 . . . . . . . 8 (𝑀...𝑗) ⊆ (ℤ𝑀)
109, 3sseqtrri 4014 . . . . . . 7 (𝑀...𝑗) ⊆ 𝑍
11 ssralv 4045 . . . . . . 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 13973 . . . . . . . 8 (𝑀...𝑗) ∈ Fin
14 fimaxre3 12193 . . . . . . . 8 (((𝑀...𝑗) ∈ Fin ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥)
1513, 14mpan 688 . . . . . . 7 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥)
16 peano2re 11419 . . . . . . . . . 10 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
1716adantl 480 . . . . . . . . 9 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 + 1) ∈ ℝ)
18 ltp1 12087 . . . . . . . . . . . . . . 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 11336 . . . . . . . . . . . . . . 15 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑥 + 1) ∈ ℝ) → (((abs‘(𝐹𝑘)) ≤ 𝑥𝑥 < (𝑥 + 1)) → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2220, 21mpd3an3 1458 . . . . . . . . . . . . . 14 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘(𝐹𝑘)) ≤ 𝑥𝑥 < (𝑥 + 1)) → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2319, 22mpan2d 692 . . . . . . . . . . . . 13 (((abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
2423expcom 412 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → ((abs‘(𝐹𝑘)) ∈ ℝ → ((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1))))
2524ralimdv 3158 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1))))
2625impcom 406 . . . . . . . . . 10 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ∀𝑘 ∈ (𝑀...𝑗)((abs‘(𝐹𝑘)) ≤ 𝑥 → (abs‘(𝐹𝑘)) < (𝑥 + 1)))
27 ralim 3075 . . . . . . . . . 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 5209 . . . . . . . . 9 (((𝑥 + 1) ∈ ℝ ∧ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < (𝑥 + 1)) → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤)
3017, 28, 29syl6an 682 . . . . . . . 8 ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) ≤ 𝑥 → ∃𝑤 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤))
3130rexlimdva 3144 . . . . . . 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 13199 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤))
35343adant3 1129 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤))
36 simp3 1135 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (abs‘(𝐹𝑘)) ∈ ℝ)
37 simp1 1133 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑤 ∈ ℝ)
38 ifcl 4575 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ℝ ∧ 𝑤 ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
3938ancoms 457 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
40393adant3 1129 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ)
41 ltletr 11338 . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑤𝑤 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
4335, 42mpan2d 692 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → ((abs‘(𝐹𝑘)) < 𝑤 → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
44 max2 13201 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤))
45443adant3 1129 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤))
46 simp2 1134 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → 𝑧 ∈ ℝ)
47 ltletr 11338 . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑧𝑧 ≤ if(𝑤𝑧, 𝑧, 𝑤)) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
4945, 48mpan2d 692 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → ((abs‘(𝐹𝑘)) < 𝑧 → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
5043, 49jaod 857 . . . . . . . . . . . . . . 15 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (abs‘(𝐹𝑘)) ∈ ℝ) → (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)))
51503expia 1118 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((abs‘(𝐹𝑘)) ∈ ℝ → (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤))))
5251ralimdv 3158 . . . . . . . . . . . . 13 ((𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (∀𝑘𝑍 (abs‘(𝐹𝑘)) ∈ ℝ → ∀𝑘𝑍 (((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) → (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤))))
53 ralim 3075 . . . . . . . . . . . . 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 5209 . . . . . . . . . . . . . 14 ((if(𝑤𝑧, 𝑧, 𝑤) ∈ ℝ ∧ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < if(𝑤𝑧, 𝑧, 𝑤)) → ∃𝑦 ∈ ℝ ∀𝑘𝑍 (abs‘(𝐹𝑘)) < 𝑦)
5655ex 411 . . . . . . . . . . . . 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 12876 . . . . . . . . . . . . . . . . . . . . . 22 (ℤ𝑀) ⊆ ℤ
603, 59eqsstri 4011 . . . . . . . . . . . . . . . . . . . . 21 𝑍 ⊆ ℤ
6160sseli 3972 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝑍𝑘 ∈ ℤ)
6260sseli 3972 . . . . . . . . . . . . . . . . . . . 20 (𝑗𝑍𝑗 ∈ ℤ)
63 uztric 12879 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗)))
6461, 62, 63syl2anr 595 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑍𝑘𝑍) → (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗)))
65 simpr 483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗𝑍𝑘𝑍) → 𝑘𝑍)
6665, 3eleqtrdi 2835 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗𝑍𝑘𝑍) → 𝑘 ∈ (ℤ𝑀))
67 elfzuzb 13530 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (𝑀...𝑗) ↔ (𝑘 ∈ (ℤ𝑀) ∧ 𝑗 ∈ (ℤ𝑘)))
6867baib 534 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (ℤ𝑀) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ𝑘)))
6966, 68syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑗𝑍𝑘𝑍) → (𝑘 ∈ (𝑀...𝑗) ↔ 𝑗 ∈ (ℤ𝑘)))
7069orbi1d 914 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑍𝑘𝑍) → ((𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗)) ↔ (𝑗 ∈ (ℤ𝑘) ∨ 𝑘 ∈ (ℤ𝑗))))
7164, 70mpbird 256 . . . . . . . . . . . . . . . . . 18 ((𝑗𝑍𝑘𝑍) → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗)))
7271ex 411 . . . . . . . . . . . . . . . . 17 (𝑗𝑍 → (𝑘𝑍 → (𝑘 ∈ (𝑀...𝑗) ∨ 𝑘 ∈ (ℤ𝑗))))
73 pm3.48 961 . . . . . . . . . . . . . . . . 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 1911 . . . . . . . . . . . . . . 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 3051 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ↔ ∀𝑘(𝑘 ∈ (𝑀...𝑗) → (abs‘(𝐹𝑘)) < 𝑤))
77 df-ral 3051 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧 ↔ ∀𝑘(𝑘 ∈ (ℤ𝑗) → (abs‘(𝐹𝑘)) < 𝑧))
7876, 77anbi12i 626 . . . . . . . . . . . . . . . 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 1865 . . . . . . . . . . . . . . . 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 277 . . . . . . . . . . . . . . 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 3051 . . . . . . . . . . . . . . 15 (∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧) ↔ ∀𝑘(𝑘𝑍 → ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧)))
8275, 80, 813imtr4g 295 . . . . . . . . . . . . . 14 (𝑗𝑍 → ((∀𝑘 ∈ (𝑀...𝑗)(abs‘(𝐹𝑘)) < 𝑤 ∧ ∀𝑘 ∈ (ℤ𝑗)(abs‘(𝐹𝑘)) < 𝑧) → ∀𝑘𝑍 ((abs‘(𝐹𝑘)) < 𝑤 ∨ (abs‘(𝐹𝑘)) < 𝑧)))
83823impib 1113 . . . . . . . . . . . . 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 1350 . . . . . . . . . . 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 452 . . . . . . . 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 3142 . . . . 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 3200 . . 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 405 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 205  wa 394  wo 845  w3a 1084  wal 1531   = wceq 1533  wcel 2098  wral 3050  wrex 3059  wss 3944  ifcif 4530   class class class wbr 5149  cfv 6549  (class class class)co 7419  Fincfn 8964  cc 11138  cr 11139  1c1 11141   + caddc 11143   < clt 11280  cle 11281  cmin 11476  cz 12591  cuz 12855  +crp 13009  ...cfz 13519  abscabs 15217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-fz 13520  df-seq 14003  df-exp 14063  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219
This theorem is referenced by:  climbdd  15654
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