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Theorem rlim 15204
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 15202 . . . . 5 Rel ⇝𝑟
21brrelex2i 5644 . . . 4 (𝐹𝑟 𝐶𝐶 ∈ V)
32a1i 11 . . 3 (𝜑 → (𝐹𝑟 𝐶𝐶 ∈ V))
4 elex 3450 . . . . 5 (𝐶 ∈ ℂ → 𝐶 ∈ V)
54ad2antrl 725 . . . 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 10952 . . . . . 6 ℂ ∈ V
10 reex 10962 . . . . . 6 ℝ ∈ V
11 elpm2r 8633 . . . . . 6 (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑pm ℝ))
129, 10, 11mpanl12 699 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑pm ℝ))
137, 8, 12syl2anc 584 . . . 4 (𝜑𝐹 ∈ (ℂ ↑pm ℝ))
14 eleq1 2826 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 ∈ (ℂ ↑pm ℝ) ↔ 𝐹 ∈ (ℂ ↑pm ℝ)))
15 eleq1 2826 . . . . . . . . 9 (𝑤 = 𝐶 → (𝑤 ∈ ℂ ↔ 𝐶 ∈ ℂ))
1614, 15bi2anan9 636 . . . . . . . 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 5814 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑤 = 𝐶) → dom 𝑓 = dom 𝐹)
19 fveq1 6773 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (𝑓𝑧) = (𝐹𝑧))
20 oveq12 7284 . . . . . . . . . . . . . . 15 (((𝑓𝑧) = (𝐹𝑧) ∧ 𝑤 = 𝐶) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐶))
2119, 20sylan 580 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑤 = 𝐶) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐶))
2221fveq2d 6778 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑤 = 𝐶) → (abs‘((𝑓𝑧) − 𝑤)) = (abs‘((𝐹𝑧) − 𝐶)))
2322breq1d 5084 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑤 = 𝐶) → ((abs‘((𝑓𝑧) − 𝑤)) < 𝑥 ↔ (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))
2423imbi2d 341 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑤 = 𝐶) → ((𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ (𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
2518, 24raleqbidv 3336 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝐶) → (∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
2625rexbidv 3226 . . . . . . . . 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 3112 . . . . . . . 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 631 . . . . . . 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 15198 . . . . . . 7 𝑟 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥))}
3028, 29brabga 5447 . . . . . 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, 31bitrdi 287 . . . . 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 6611 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐴)
3837raleqdv 3348 . . . . . 6 (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
39 rlim.4 . . . . . . . . . 10 ((𝜑𝑧𝐴) → (𝐹𝑧) = 𝐵)
4039fvoveq1d 7297 . . . . . . . . 9 ((𝜑𝑧𝐴) → (abs‘((𝐹𝑧) − 𝐶)) = (abs‘(𝐵𝐶)))
4140breq1d 5084 . . . . . . . 8 ((𝜑𝑧𝐴) → ((abs‘((𝐹𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵𝐶)) < 𝑥))
4241imbi2d 341 . . . . . . 7 ((𝜑𝑧𝐴) → ((𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4342ralbidva 3111 . . . . . 6 (𝜑 → (∀𝑧𝐴 (𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4438, 43bitrd 278 . . . . 5 (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4544rexbidv 3226 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4645ralbidv 3112 . . 3 (𝜑 → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4746anbi2d 629 . 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 307 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 205  wa 396   = wceq 1539  wcel 2106  wral 3064  wrex 3065  Vcvv 3432  wss 3887   class class class wbr 5074  dom cdm 5589  wf 6429  cfv 6433  (class class class)co 7275  pm cpm 8616  cc 10869  cr 10870   < clt 11009  cle 11010  cmin 11205  +crp 12730  abscabs 14945  𝑟 crli 15194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-pm 8618  df-rlim 15198
This theorem is referenced by:  rlim2  15205  rlimcl  15212  rlimclim  15255  rlimres  15267  caurcvgr  15385
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