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Theorem rlim 14840
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 14838 . . . . 5 Rel ⇝𝑟
21brrelex2i 5602 . . . 4 (𝐹𝑟 𝐶𝐶 ∈ V)
32a1i 11 . . 3 (𝜑 → (𝐹𝑟 𝐶𝐶 ∈ V))
4 elex 3510 . . . . 5 (𝐶 ∈ ℂ → 𝐶 ∈ V)
54ad2antrl 724 . . . 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 10606 . . . . . 6 ℂ ∈ V
10 reex 10616 . . . . . 6 ℝ ∈ V
11 elpm2r 8413 . . . . . 6 (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑pm ℝ))
129, 10, 11mpanl12 698 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑pm ℝ))
137, 8, 12syl2anc 584 . . . 4 (𝜑𝐹 ∈ (ℂ ↑pm ℝ))
14 eleq1 2897 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 ∈ (ℂ ↑pm ℝ) ↔ 𝐹 ∈ (ℂ ↑pm ℝ)))
15 eleq1 2897 . . . . . . . . 9 (𝑤 = 𝐶 → (𝑤 ∈ ℂ ↔ 𝐶 ∈ ℂ))
1614, 15bi2anan9 635 . . . . . . . 8 ((𝑓 = 𝐹𝑤 = 𝐶) → ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ ℂ)))
17 simpl 483 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑤 = 𝐶) → 𝑓 = 𝐹)
1817dmeqd 5767 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑤 = 𝐶) → dom 𝑓 = dom 𝐹)
19 fveq1 6662 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (𝑓𝑧) = (𝐹𝑧))
20 oveq12 7154 . . . . . . . . . . . . . . 15 (((𝑓𝑧) = (𝐹𝑧) ∧ 𝑤 = 𝐶) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐶))
2119, 20sylan 580 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑤 = 𝐶) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐶))
2221fveq2d 6667 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑤 = 𝐶) → (abs‘((𝑓𝑧) − 𝑤)) = (abs‘((𝐹𝑧) − 𝐶)))
2322breq1d 5067 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑤 = 𝐶) → ((abs‘((𝑓𝑧) − 𝑤)) < 𝑥 ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
2423imbi2d 342 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑤 = 𝐶) → ((𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ (𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
2518, 24raleqbidv 3399 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝐶) → (∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
2625rexbidv 3294 . . . . . . . . 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 3194 . . . . . . . 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 630 . . . . . . 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 14834 . . . . . . 7 𝑟 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥))}
3028, 29brabga 5412 . . . . . 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 469 . . . . . 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 288 . . . . 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 413 . . . 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 381 . 2 (𝜑 → (𝐹𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))))
3613biantrurd 533 . 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)))))
377fdmd 6516 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐴)
3837raleqdv 3413 . . . . . 6 (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
39 rlim.4 . . . . . . . . . 10 ((𝜑𝑧𝐴) → (𝐹𝑧) = 𝐵)
4039fvoveq1d 7167 . . . . . . . . 9 ((𝜑𝑧𝐴) → (abs‘((𝐹𝑧) − 𝐶)) = (abs‘(𝐵𝐶)))
4140breq1d 5067 . . . . . . . 8 ((𝜑𝑧𝐴) → ((abs‘((𝐹𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵𝐶)) < 𝑥))
4241imbi2d 342 . . . . . . 7 ((𝜑𝑧𝐴) → ((𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4342ralbidva 3193 . . . . . 6 (𝜑 → (∀𝑧𝐴 (𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4438, 43bitrd 280 . . . . 5 (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4544rexbidv 3294 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4645ralbidv 3194 . . 3 (𝜑 → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4746anbi2d 628 . 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))))
4835, 36, 473bitr2d 308 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 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  wss 3933   class class class wbr 5057  dom cdm 5548  wf 6344  cfv 6348  (class class class)co 7145  pm cpm 8396  cc 10523  cr 10524   < clt 10663  cle 10664  cmin 10858  +crp 12377  abscabs 14581  𝑟 crli 14830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-pm 8398  df-rlim 14834
This theorem is referenced by:  rlim2  14841  rlimcl  14848  rlimclim  14891  rlimres  14903  caurcvgr  15018
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