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Theorem rlim2 14512
Description: Rewrite rlim 14511 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 2765 . . . . 5 (𝑧𝐴𝐵) = (𝑧𝐴𝐵)
32fmpt 6570 . . . 4 (∀𝑧𝐴 𝐵 ∈ ℂ ↔ (𝑧𝐴𝐵):𝐴⟶ℂ)
41, 3sylib 209 . . 3 (𝜑 → (𝑧𝐴𝐵):𝐴⟶ℂ)
5 rlim2.2 . . 3 (𝜑𝐴 ⊆ ℝ)
6 eqidd 2766 . . 3 ((𝜑𝑤𝐴) → ((𝑧𝐴𝐵)‘𝑤) = ((𝑧𝐴𝐵)‘𝑤))
74, 5, 6rlim 14511 . 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 2009 . . . . . . 7 𝑧 𝑦𝑤
11 nfcv 2907 . . . . . . . . 9 𝑧abs
12 nffvmpt1 6386 . . . . . . . . . 10 𝑧((𝑧𝐴𝐵)‘𝑤)
13 nfcv 2907 . . . . . . . . . 10 𝑧
14 nfcv 2907 . . . . . . . . . 10 𝑧𝐶
1512, 13, 14nfov 6872 . . . . . . . . 9 𝑧(((𝑧𝐴𝐵)‘𝑤) − 𝐶)
1611, 15nffv 6385 . . . . . . . 8 𝑧(abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶))
17 nfcv 2907 . . . . . . . 8 𝑧 <
18 nfcv 2907 . . . . . . . 8 𝑧𝑥
1916, 17, 18nfbr 4856 . . . . . . 7 𝑧(abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥
2010, 19nfim 1995 . . . . . 6 𝑧(𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥)
21 nfv 2009 . . . . . 6 𝑤(𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥)
22 breq2 4813 . . . . . . 7 (𝑤 = 𝑧 → (𝑦𝑤𝑦𝑧))
2322imbrov2fvoveq 6867 . . . . . 6 (𝑤 = 𝑧 → ((𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥)))
2420, 21, 23cbvral 3315 . . . . 5 (∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥))
252fvmpt2 6480 . . . . . . . . . 10 ((𝑧𝐴𝐵 ∈ ℂ) → ((𝑧𝐴𝐵)‘𝑧) = 𝐵)
2625fvoveq1d 6864 . . . . . . . . 9 ((𝑧𝐴𝐵 ∈ ℂ) → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) = (abs‘(𝐵𝐶)))
2726breq1d 4819 . . . . . . . 8 ((𝑧𝐴𝐵 ∈ ℂ) → ((abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵𝐶)) < 𝑥))
2827imbi2d 331 . . . . . . 7 ((𝑧𝐴𝐵 ∈ ℂ) → ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
2928ralimiaa 3098 . . . . . 6 (∀𝑧𝐴 𝐵 ∈ ℂ → ∀𝑧𝐴 ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
30 ralbi 3215 . . . . . 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 274 . . . 4 (𝜑 → (∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3332rexbidv 3199 . . 3 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3433ralbidv 3133 . 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 298 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 197  wa 384  wcel 2155  wral 3055  wrex 3056  wss 3732   class class class wbr 4809  cmpt 4888  wf 6064  cfv 6068  (class class class)co 6842  cc 10187  cr 10188   < clt 10328  cle 10329  cmin 10520  +crp 12028  abscabs 14259  𝑟 crli 14501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-pm 8063  df-rlim 14505
This theorem is referenced by:  rlim2lt  14513  rlim3  14514  rlim0  14524  rlimi  14529  rlimconst  14560  climrlim2  14563  rlimcn1  14604  rlimcn2  14606  chtppilim  25455  pntlem3  25589
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