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Theorem rlim2 14271
Description: Rewrite rlim 14270 for a mapping operation. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
Hypotheses
Ref Expression
rlim2.1 (𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)
rlim2.2 (𝜑𝐴 ⊆ ℝ)
rlim2.3 (𝜑𝐶 ∈ ℂ)
Assertion
Ref Expression
rlim2 (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧)   𝐵(𝑧)

Proof of Theorem rlim2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 rlim2.1 . . . 4 (𝜑 → ∀𝑧𝐴 𝐵 ∈ ℂ)
2 eqid 2651 . . . . 5 (𝑧𝐴𝐵) = (𝑧𝐴𝐵)
32fmpt 6421 . . . 4 (∀𝑧𝐴 𝐵 ∈ ℂ ↔ (𝑧𝐴𝐵):𝐴⟶ℂ)
41, 3sylib 208 . . 3 (𝜑 → (𝑧𝐴𝐵):𝐴⟶ℂ)
5 rlim2.2 . . 3 (𝜑𝐴 ⊆ ℝ)
6 eqidd 2652 . . 3 ((𝜑𝑤𝐴) → ((𝑧𝐴𝐵)‘𝑤) = ((𝑧𝐴𝐵)‘𝑤))
74, 5, 6rlim 14270 . 2 (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 𝐶 ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥))))
8 rlim2.3 . . 3 (𝜑𝐶 ∈ ℂ)
98biantrurd 528 . 2 (𝜑 → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥))))
10 nfv 1883 . . . . . . 7 𝑧 𝑦𝑤
11 nfcv 2793 . . . . . . . . 9 𝑧abs
12 nffvmpt1 6237 . . . . . . . . . 10 𝑧((𝑧𝐴𝐵)‘𝑤)
13 nfcv 2793 . . . . . . . . . 10 𝑧
14 nfcv 2793 . . . . . . . . . 10 𝑧𝐶
1512, 13, 14nfov 6716 . . . . . . . . 9 𝑧(((𝑧𝐴𝐵)‘𝑤) − 𝐶)
1611, 15nffv 6236 . . . . . . . 8 𝑧(abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶))
17 nfcv 2793 . . . . . . . 8 𝑧 <
18 nfcv 2793 . . . . . . . 8 𝑧𝑥
1916, 17, 18nfbr 4732 . . . . . . 7 𝑧(abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥
2010, 19nfim 1865 . . . . . 6 𝑧(𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥)
21 nfv 1883 . . . . . 6 𝑤(𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥)
22 breq2 4689 . . . . . . 7 (𝑤 = 𝑧 → (𝑦𝑤𝑦𝑧))
23 fveq2 6229 . . . . . . . . . 10 (𝑤 = 𝑧 → ((𝑧𝐴𝐵)‘𝑤) = ((𝑧𝐴𝐵)‘𝑧))
2423oveq1d 6705 . . . . . . . . 9 (𝑤 = 𝑧 → (((𝑧𝐴𝐵)‘𝑤) − 𝐶) = (((𝑧𝐴𝐵)‘𝑧) − 𝐶))
2524fveq2d 6233 . . . . . . . 8 (𝑤 = 𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) = (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)))
2625breq1d 4695 . . . . . . 7 (𝑤 = 𝑧 → ((abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥 ↔ (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥))
2722, 26imbi12d 333 . . . . . 6 (𝑤 = 𝑧 → ((𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥)))
2820, 21, 27cbvral 3197 . . . . 5 (∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥))
292fvmpt2 6330 . . . . . . . . . . 11 ((𝑧𝐴𝐵 ∈ ℂ) → ((𝑧𝐴𝐵)‘𝑧) = 𝐵)
3029oveq1d 6705 . . . . . . . . . 10 ((𝑧𝐴𝐵 ∈ ℂ) → (((𝑧𝐴𝐵)‘𝑧) − 𝐶) = (𝐵𝐶))
3130fveq2d 6233 . . . . . . . . 9 ((𝑧𝐴𝐵 ∈ ℂ) → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) = (abs‘(𝐵𝐶)))
3231breq1d 4695 . . . . . . . 8 ((𝑧𝐴𝐵 ∈ ℂ) → ((abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵𝐶)) < 𝑥))
3332imbi2d 329 . . . . . . 7 ((𝑧𝐴𝐵 ∈ ℂ) → ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3433ralimiaa 2980 . . . . . 6 (∀𝑧𝐴 𝐵 ∈ ℂ → ∀𝑧𝐴 ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
35 ralbi 3097 . . . . . 6 (∀𝑧𝐴 ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)) → (∀𝑧𝐴 (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
361, 34, 353syl 18 . . . . 5 (𝜑 → (∀𝑧𝐴 (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3728, 36syl5bb 272 . . . 4 (𝜑 → (∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3837rexbidv 3081 . . 3 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3938ralbidv 3015 . 2 (𝜑 → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
407, 9, 393bitr2d 296 1 (𝜑 → ((𝑧𝐴𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2030  wral 2941  wrex 2942  wss 3607   class class class wbr 4685  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  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-csb 3567  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-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  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:  rlim2lt  14272  rlim3  14273  rlim0  14283  rlimi  14288  rlimconst  14319  climrlim2  14322  rlimcn1  14363  rlimcn2  14365  chtppilim  25209  pntlem3  25343
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