Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rlim Structured version   Visualization version   GIF version

Theorem rlim 14842
 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 14840 . . . . 5 Rel ⇝𝑟
21brrelex2i 5608 . . . 4 (𝐹𝑟 𝐶𝐶 ∈ V)
32a1i 11 . . 3 (𝜑 → (𝐹𝑟 𝐶𝐶 ∈ V))
4 elex 3518 . . . . 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 10607 . . . . . 6 ℂ ∈ V
10 reex 10617 . . . . . 6 ℝ ∈ V
11 elpm2r 8414 . . . . . 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 2905 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 ∈ (ℂ ↑pm ℝ) ↔ 𝐹 ∈ (ℂ ↑pm ℝ)))
15 eleq1 2905 . . . . . . . . 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 5773 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑤 = 𝐶) → dom 𝑓 = dom 𝐹)
19 fveq1 6666 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (𝑓𝑧) = (𝐹𝑧))
20 oveq12 7157 . . . . . . . . . . . . . . 15 (((𝑓𝑧) = (𝐹𝑧) ∧ 𝑤 = 𝐶) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐶))
2119, 20sylan 580 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑤 = 𝐶) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐶))
2221fveq2d 6671 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑤 = 𝐶) → (abs‘((𝑓𝑧) − 𝑤)) = (abs‘((𝐹𝑧) − 𝐶)))
2322breq1d 5073 . . . . . . . . . . . 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 3407 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝐶) → (∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
2625rexbidv 3302 . . . . . . . . 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 3202 . . . . . . . 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 14836 . . . . . . 7 𝑟 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥))}
3028, 29brabga 5418 . . . . . 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 6520 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐴)
3837raleqdv 3421 . . . . . 6 (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
39 rlim.4 . . . . . . . . . 10 ((𝜑𝑧𝐴) → (𝐹𝑧) = 𝐵)
4039fvoveq1d 7170 . . . . . . . . 9 ((𝜑𝑧𝐴) → (abs‘((𝐹𝑧) − 𝐶)) = (abs‘(𝐵𝐶)))
4140breq1d 5073 . . . . . . . 8 ((𝜑𝑧𝐴) → ((abs‘((𝐹𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵𝐶)) < 𝑥))
4241imbi2d 342 . . . . . . 7 ((𝜑𝑧𝐴) → ((𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4342ralbidva 3201 . . . . . 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 3302 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4645ralbidv 3202 . . 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 1530   ∈ wcel 2107  ∀wral 3143  ∃wrex 3144  Vcvv 3500   ⊆ wss 3940   class class class wbr 5063  dom cdm 5554  ⟶wf 6348  ‘cfv 6352  (class class class)co 7148   ↑pm cpm 8397  ℂcc 10524  ℝcr 10525   < clt 10664   ≤ cle 10665   − cmin 10859  ℝ+crp 12379  abscabs 14583   ⇝𝑟 crli 14832 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-pm 8399  df-rlim 14836 This theorem is referenced by:  rlim2  14843  rlimcl  14850  rlimclim  14893  rlimres  14905  caurcvgr  15020
 Copyright terms: Public domain W3C validator