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Theorem rlim2 14848
Description: Rewrite rlim 14847 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 2826 . . . . 5 (𝑧𝐴𝐵) = (𝑧𝐴𝐵)
32fmpt 6872 . . . 4 (∀𝑧𝐴 𝐵 ∈ ℂ ↔ (𝑧𝐴𝐵):𝐴⟶ℂ)
41, 3sylib 219 . . 3 (𝜑 → (𝑧𝐴𝐵):𝐴⟶ℂ)
5 rlim2.2 . . 3 (𝜑𝐴 ⊆ ℝ)
6 eqidd 2827 . . 3 ((𝜑𝑤𝐴) → ((𝑧𝐴𝐵)‘𝑤) = ((𝑧𝐴𝐵)‘𝑤))
74, 5, 6rlim 14847 . 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 533 . 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 1908 . . . . . . 7 𝑧 𝑦𝑤
11 nfcv 2982 . . . . . . . . 9 𝑧abs
12 nffvmpt1 6680 . . . . . . . . . 10 𝑧((𝑧𝐴𝐵)‘𝑤)
13 nfcv 2982 . . . . . . . . . 10 𝑧
14 nfcv 2982 . . . . . . . . . 10 𝑧𝐶
1512, 13, 14nfov 7180 . . . . . . . . 9 𝑧(((𝑧𝐴𝐵)‘𝑤) − 𝐶)
1611, 15nffv 6679 . . . . . . . 8 𝑧(abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶))
17 nfcv 2982 . . . . . . . 8 𝑧 <
18 nfcv 2982 . . . . . . . 8 𝑧𝑥
1916, 17, 18nfbr 5110 . . . . . . 7 𝑧(abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥
2010, 19nfim 1890 . . . . . 6 𝑧(𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥)
21 nfv 1908 . . . . . 6 𝑤(𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥)
22 breq2 5067 . . . . . . 7 (𝑤 = 𝑧 → (𝑦𝑤𝑦𝑧))
2322imbrov2fvoveq 7175 . . . . . 6 (𝑤 = 𝑧 → ((𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥)))
2420, 21, 23cbvralw 3447 . . . . 5 (∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥))
252fvmpt2 6777 . . . . . . . . . 10 ((𝑧𝐴𝐵 ∈ ℂ) → ((𝑧𝐴𝐵)‘𝑧) = 𝐵)
2625fvoveq1d 7172 . . . . . . . . 9 ((𝑧𝐴𝐵 ∈ ℂ) → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) = (abs‘(𝐵𝐶)))
2726breq1d 5073 . . . . . . . 8 ((𝑧𝐴𝐵 ∈ ℂ) → ((abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵𝐶)) < 𝑥))
2827imbi2d 342 . . . . . . 7 ((𝑧𝐴𝐵 ∈ ℂ) → ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
2928ralimiaa 3164 . . . . . 6 (∀𝑧𝐴 𝐵 ∈ ℂ → ∀𝑧𝐴 ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
30 ralbi 3172 . . . . . 6 (∀𝑧𝐴 ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)) → (∀𝑧𝐴 (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
311, 29, 303syl 18 . . . . 5 (𝜑 → (∀𝑧𝐴 (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3224, 31syl5bb 284 . . . 4 (𝜑 → (∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3332rexbidv 3302 . . 3 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3433ralbidv 3202 . 2 (𝜑 → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
357, 9, 343bitr2d 308 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 207  wa 396  wcel 2107  wral 3143  wrex 3144  wss 3940   class class class wbr 5063  cmpt 5143  wf 6350  cfv 6354  (class class class)co 7150  cc 10529  cr 10530   < clt 10669  cle 10670  cmin 10864  +crp 12384  abscabs 14588  𝑟 crli 14837
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 7455  ax-cnex 10587  ax-resscn 10588
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-csb 3888  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-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-pm 8404  df-rlim 14841
This theorem is referenced by:  rlim2lt  14849  rlim3  14850  rlim0  14860  rlimi  14865  rlimconst  14896  climrlim2  14899  rlimcn1  14940  rlimcn2  14942  chtppilim  25984  pntlem3  26118
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