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Theorem rlim2 15427
Description: Rewrite rlim 15426 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 2733 . . . . 5 (𝑧𝐴𝐵) = (𝑧𝐴𝐵)
32fmpt 7097 . . . 4 (∀𝑧𝐴 𝐵 ∈ ℂ ↔ (𝑧𝐴𝐵):𝐴⟶ℂ)
41, 3sylib 217 . . 3 (𝜑 → (𝑧𝐴𝐵):𝐴⟶ℂ)
5 rlim2.2 . . 3 (𝜑𝐴 ⊆ ℝ)
6 eqidd 2734 . . 3 ((𝜑𝑤𝐴) → ((𝑧𝐴𝐵)‘𝑤) = ((𝑧𝐴𝐵)‘𝑤))
74, 5, 6rlim 15426 . 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 534 . 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 1918 . . . . . . 7 𝑧 𝑦𝑤
11 nfcv 2904 . . . . . . . . 9 𝑧abs
12 nffvmpt1 6892 . . . . . . . . . 10 𝑧((𝑧𝐴𝐵)‘𝑤)
13 nfcv 2904 . . . . . . . . . 10 𝑧
14 nfcv 2904 . . . . . . . . . 10 𝑧𝐶
1512, 13, 14nfov 7426 . . . . . . . . 9 𝑧(((𝑧𝐴𝐵)‘𝑤) − 𝐶)
1611, 15nffv 6891 . . . . . . . 8 𝑧(abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶))
17 nfcv 2904 . . . . . . . 8 𝑧 <
18 nfcv 2904 . . . . . . . 8 𝑧𝑥
1916, 17, 18nfbr 5191 . . . . . . 7 𝑧(abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥
2010, 19nfim 1900 . . . . . 6 𝑧(𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥)
21 nfv 1918 . . . . . 6 𝑤(𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥)
22 breq2 5148 . . . . . . 7 (𝑤 = 𝑧 → (𝑦𝑤𝑦𝑧))
2322imbrov2fvoveq 7421 . . . . . 6 (𝑤 = 𝑧 → ((𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥)))
2420, 21, 23cbvralw 3304 . . . . 5 (∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥))
252fvmpt2 6998 . . . . . . . . . 10 ((𝑧𝐴𝐵 ∈ ℂ) → ((𝑧𝐴𝐵)‘𝑧) = 𝐵)
2625fvoveq1d 7418 . . . . . . . . 9 ((𝑧𝐴𝐵 ∈ ℂ) → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) = (abs‘(𝐵𝐶)))
2726breq1d 5154 . . . . . . . 8 ((𝑧𝐴𝐵 ∈ ℂ) → ((abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵𝐶)) < 𝑥))
2827imbi2d 341 . . . . . . 7 ((𝑧𝐴𝐵 ∈ ℂ) → ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
2928ralimiaa 3083 . . . . . 6 (∀𝑧𝐴 𝐵 ∈ ℂ → ∀𝑧𝐴 ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
30 ralbi 3104 . . . . . 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, 31bitrid 283 . . . 4 (𝜑 → (∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3332rexbidv 3179 . . 3 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3433ralbidv 3178 . 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 307 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 205  wa 397  wcel 2107  wral 3062  wrex 3071  wss 3946   class class class wbr 5144  cmpt 5227  wf 6531  cfv 6535  (class class class)co 7396  cc 11095  cr 11096   < clt 11235  cle 11236  cmin 11431  +crp 12961  abscabs 15168  𝑟 crli 15416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-pm 8811  df-rlim 15420
This theorem is referenced by:  rlim2lt  15428  rlim3  15429  rlim0  15439  rlimi  15444  rlimconst  15475  climrlim2  15478  rlimcn1  15519  rlimcn3  15521  chtppilim  26945  pntlem3  27079
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