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Theorem rlim 14425
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 14423 . . . . 5 Rel ⇝𝑟
21brrelex2i 5316 . . . 4 (𝐹𝑟 𝐶𝐶 ∈ V)
32a1i 11 . . 3 (𝜑 → (𝐹𝑟 𝐶𝐶 ∈ V))
4 elex 3352 . . . . 5 (𝐶 ∈ ℂ → 𝐶 ∈ V)
54ad2antrl 766 . . . 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 10209 . . . . . 6 ℂ ∈ V
10 reex 10219 . . . . . 6 ℝ ∈ V
11 elpm2r 8041 . . . . . 6 (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑pm ℝ))
129, 10, 11mpanl12 720 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑pm ℝ))
137, 8, 12syl2anc 696 . . . 4 (𝜑𝐹 ∈ (ℂ ↑pm ℝ))
14 eleq1 2827 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 ∈ (ℂ ↑pm ℝ) ↔ 𝐹 ∈ (ℂ ↑pm ℝ)))
15 eleq1 2827 . . . . . . . . 9 (𝑤 = 𝐶 → (𝑤 ∈ ℂ ↔ 𝐶 ∈ ℂ))
1614, 15bi2anan9 953 . . . . . . . 8 ((𝑓 = 𝐹𝑤 = 𝐶) → ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ ℂ)))
17 simpl 474 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑤 = 𝐶) → 𝑓 = 𝐹)
1817dmeqd 5481 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑤 = 𝐶) → dom 𝑓 = dom 𝐹)
19 fveq1 6351 . . . . . . . . . . . . . . 15 (𝑓 = 𝐹 → (𝑓𝑧) = (𝐹𝑧))
20 oveq12 6822 . . . . . . . . . . . . . . 15 (((𝑓𝑧) = (𝐹𝑧) ∧ 𝑤 = 𝐶) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐶))
2119, 20sylan 489 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑤 = 𝐶) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐶))
2221fveq2d 6356 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑤 = 𝐶) → (abs‘((𝑓𝑧) − 𝑤)) = (abs‘((𝐹𝑧) − 𝐶)))
2322breq1d 4814 . . . . . . . . . . . 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 3291 . . . . . . . . . 10 ((𝑓 = 𝐹𝑤 = 𝐶) → (∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
2625rexbidv 3190 . . . . . . . . 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 3124 . . . . . . . 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 749 . . . . . . 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 14419 . . . . . . 7 𝑟 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦𝑧 → (abs‘((𝑓𝑧) − 𝑤)) < 𝑥))}
3028, 29brabga 5139 . . . . . 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 684 . . . . . 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 530 . 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 6212 . . . . . . . 8 (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴)
387, 37syl 17 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐴)
3938raleqdv 3283 . . . . . 6 (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥)))
40 rlim.4 . . . . . . . . . . 11 ((𝜑𝑧𝐴) → (𝐹𝑧) = 𝐵)
4140oveq1d 6828 . . . . . . . . . 10 ((𝜑𝑧𝐴) → ((𝐹𝑧) − 𝐶) = (𝐵𝐶))
4241fveq2d 6356 . . . . . . . . 9 ((𝜑𝑧𝐴) → (abs‘((𝐹𝑧) − 𝐶)) = (abs‘(𝐵𝐶)))
4342breq1d 4814 . . . . . . . 8 ((𝜑𝑧𝐴) → ((abs‘((𝐹𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵𝐶)) < 𝑥))
4443imbi2d 329 . . . . . . 7 ((𝜑𝑧𝐴) → ((𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4544ralbidva 3123 . . . . . 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 3190 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4847ralbidv 3124 . . 3 (𝜑 → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
4948anbi2d 742 . 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 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  wss 3715   class class class wbr 4804  dom cdm 5266  wf 6045  cfv 6049  (class class class)co 6813  pm cpm 8024  cc 10126  cr 10127   < clt 10266  cle 10267  cmin 10458  +crp 12025  abscabs 14173  𝑟 crli 14415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-pm 8026  df-rlim 14419
This theorem is referenced by:  rlim2  14426  rlimcl  14433  rlimclim  14476  rlimres  14488  caurcvgr  14603
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