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Theorem rlim 14270
Description: Express the predicate: The limit of complex number function 𝐹 is 𝐶, or 𝐹 converges to 𝐶, in the real sense. This means that for any real 𝑥, no matter how small, there always exists a number 𝑦 such that the absolute difference of any number in the function beyond 𝑦 and the limit is less than 𝑥. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypotheses
Ref Expression
rlim.1 (𝜑𝐹:𝐴⟶ℂ)
rlim.2 (𝜑𝐴 ⊆ ℝ)
rlim.4 ((𝜑𝑧𝐴) → (𝐹𝑧) = 𝐵)
Assertion
Ref Expression
rlim (𝜑 → (𝐹𝑟 𝐶 ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥))))
Distinct variable groups:   𝑧,𝐴   𝑥,𝑦,𝑧,𝐶   𝑥,𝐹,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑧)

Proof of Theorem rlim
Dummy variables 𝑤 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimrel 14268 . . . . 5 Rel ⇝𝑟
21brrelex2i 5193 . . . 4 (𝐹𝑟 𝐶𝐶 ∈ V)
32a1i 11 . . 3 (𝜑 → (𝐹𝑟 𝐶𝐶 ∈ V))
4 elex 3243 . . . . 5 (𝐶 ∈ ℂ → 𝐶 ∈ V)
54ad2antrl 764 . . . 4 ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))) → 𝐶 ∈ V)
65a1i 11 . . 3 (𝜑 → ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))) → 𝐶 ∈ V))
7 rlim.1 . . . . 5 (𝜑𝐹:𝐴⟶ℂ)
8 rlim.2 . . . . 5 (𝜑𝐴 ⊆ ℝ)
9 cnex 10055 . . . . . 6 ℂ ∈ V
10 reex 10065 . . . . . 6 ℝ ∈ V
11 elpm2r 7917 . . . . . 6 (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑pm ℝ))
129, 10, 11mpanl12 718 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑pm ℝ))
137, 8, 12syl2anc 694 . . . 4 (𝜑𝐹 ∈ (ℂ ↑pm ℝ))
14 eleq1 2718 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 ∈ (ℂ ↑pm ℝ) ↔ 𝐹 ∈ (ℂ ↑pm ℝ)))
15 eleq1 2718 . . . . . . . . 9 (𝑤 = 𝐶 → (𝑤 ∈ ℂ ↔ 𝐶 ∈ ℂ))
1614, 15bi2anan9 935 . . . . . . . 8 ((𝑓 = 𝐹𝑤 = 𝐶) → ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ ℂ)))
17 simpl 472 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑤 = 𝐶) → 𝑓 = 𝐹)
1817dmeqd 5358 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑤 = 𝐶) → dom 𝑓 = dom 𝐹)
19 fveq1 6228 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (𝑓𝑧) = (𝐹𝑧))
20 oveq12 6699 . . . . . . . . . . . . . . 15 (((𝑓𝑧) = (𝐹𝑧) ∧ 𝑤 = 𝐶) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐶))
2119, 20sylan 487 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑤 = 𝐶) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐶))
2221fveq2d 6233 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑤 = 𝐶) → (abs‘((𝑓𝑧) − 𝑤)) = (abs‘((𝐹𝑧) − 𝐶)))
2322breq1d 4695 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑤 = 𝐶) → ((abs‘((𝑓𝑧) − 𝑤)) < 𝑥 ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
2423imbi2d 329 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑤 = 𝐶) → ((𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ (𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
2518, 24raleqbidv 3182 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝐶) → (∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
2625rexbidv 3081 . . . . . . . . 9 ((𝑓 = 𝐹𝑤 = 𝐶) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
2726ralbidv 3015 . . . . . . . 8 ((𝑓 = 𝐹𝑤 = 𝐶) → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
2816, 27anbi12d 747 . . . . . . 7 ((𝑓 = 𝐹𝑤 = 𝐶) → (((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥)) ↔ ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
29 df-rlim 14264 . . . . . . 7 𝑟 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥))}
3028, 29brabga 5018 . . . . . 6 ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ V) → (𝐹𝑟 𝐶 ↔ ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
31 anass 682 . . . . . 6 (((𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))
3230, 31syl6bb 276 . . . . 5 ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ V) → (𝐹𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))))
3332ex 449 . . . 4 (𝐹 ∈ (ℂ ↑pm ℝ) → (𝐶 ∈ V → (𝐹𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))))
3413, 33syl 17 . . 3 (𝜑 → (𝐶 ∈ V → (𝐹𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))))
353, 6, 34pm5.21ndd 368 . 2 (𝜑 → (𝐹𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))))
3613biantrurd 528 . 2 (𝜑 → ((𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))))
37 fdm 6089 . . . . . . . 8 (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴)
387, 37syl 17 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐴)
3938raleqdv 3174 . . . . . 6 (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
40 rlim.4 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → (𝐹𝑧) = 𝐵)
4140oveq1d 6705 . . . . . . . . . 10 ((𝜑𝑧𝐴) → ((𝐹𝑧) − 𝐶) = (𝐵𝐶))
4241fveq2d 6233 . . . . . . . . 9 ((𝜑𝑧𝐴) → (abs‘((𝐹𝑧) − 𝐶)) = (abs‘(𝐵𝐶)))
4342breq1d 4695 . . . . . . . 8 ((𝜑𝑧𝐴) → ((abs‘((𝐹𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵𝐶)) < 𝑥))
4443imbi2d 329 . . . . . . 7 ((𝜑𝑧𝐴) → ((𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4544ralbidva 3014 . . . . . 6 (𝜑 → (∀𝑧𝐴 (𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4639, 45bitrd 268 . . . . 5 (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4746rexbidv 3081 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4847ralbidv 3015 . . 3 (𝜑 → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4948anbi2d 740 . 2 (𝜑 → ((𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥))))
5035, 36, 493bitr2d 296 1 (𝜑 → (𝐹𝑟 𝐶 ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607   class class class wbr 4685  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  pm cpm 7900  cc 9972  cr 9973   < clt 10112  cle 10113  cmin 10304  +crp 11870  abscabs 14018  𝑟 crli 14260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-pm 7902  df-rlim 14264
This theorem is referenced by:  rlim2  14271  rlimcl  14278  rlimclim  14321  rlimres  14333  caurcvgr  14448
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