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Theorem rlim2 14428
Description: Rewrite rlim 14427 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 2771 . . . . 5 (𝑧𝐴𝐵) = (𝑧𝐴𝐵)
32fmpt 6521 . . . 4 (∀𝑧𝐴 𝐵 ∈ ℂ ↔ (𝑧𝐴𝐵):𝐴⟶ℂ)
41, 3sylib 208 . . 3 (𝜑 → (𝑧𝐴𝐵):𝐴⟶ℂ)
5 rlim2.2 . . 3 (𝜑𝐴 ⊆ ℝ)
6 eqidd 2772 . . 3 ((𝜑𝑤𝐴) → ((𝑧𝐴𝐵)‘𝑤) = ((𝑧𝐴𝐵)‘𝑤))
74, 5, 6rlim 14427 . 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 522 . 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 1995 . . . . . . 7 𝑧 𝑦𝑤
11 nfcv 2913 . . . . . . . . 9 𝑧abs
12 nffvmpt1 6338 . . . . . . . . . 10 𝑧((𝑧𝐴𝐵)‘𝑤)
13 nfcv 2913 . . . . . . . . . 10 𝑧
14 nfcv 2913 . . . . . . . . . 10 𝑧𝐶
1512, 13, 14nfov 6819 . . . . . . . . 9 𝑧(((𝑧𝐴𝐵)‘𝑤) − 𝐶)
1611, 15nffv 6337 . . . . . . . 8 𝑧(abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶))
17 nfcv 2913 . . . . . . . 8 𝑧 <
18 nfcv 2913 . . . . . . . 8 𝑧𝑥
1916, 17, 18nfbr 4833 . . . . . . 7 𝑧(abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥
2010, 19nfim 1977 . . . . . 6 𝑧(𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥)
21 nfv 1995 . . . . . 6 𝑤(𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥)
22 breq2 4790 . . . . . . 7 (𝑤 = 𝑧 → (𝑦𝑤𝑦𝑧))
23 fveq2 6330 . . . . . . . . 9 (𝑤 = 𝑧 → ((𝑧𝐴𝐵)‘𝑤) = ((𝑧𝐴𝐵)‘𝑧))
2423fvoveq1d 6813 . . . . . . . 8 (𝑤 = 𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) = (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)))
2524breq1d 4796 . . . . . . 7 (𝑤 = 𝑧 → ((abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥 ↔ (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥))
2622, 25imbi12d 333 . . . . . 6 (𝑤 = 𝑧 → ((𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥)))
2720, 21, 26cbvral 3316 . . . . 5 (∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥))
282fvmpt2 6431 . . . . . . . . . 10 ((𝑧𝐴𝐵 ∈ ℂ) → ((𝑧𝐴𝐵)‘𝑧) = 𝐵)
2928fvoveq1d 6813 . . . . . . . . 9 ((𝑧𝐴𝐵 ∈ ℂ) → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) = (abs‘(𝐵𝐶)))
3029breq1d 4796 . . . . . . . 8 ((𝑧𝐴𝐵 ∈ ℂ) → ((abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵𝐶)) < 𝑥))
3130imbi2d 329 . . . . . . 7 ((𝑧𝐴𝐵 ∈ ℂ) → ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3231ralimiaa 3100 . . . . . 6 (∀𝑧𝐴 𝐵 ∈ ℂ → ∀𝑧𝐴 ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
33 ralbi 3216 . . . . . 6 (∀𝑧𝐴 ((𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)) → (∀𝑧𝐴 (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
341, 32, 333syl 18 . . . . 5 (𝜑 → (∀𝑧𝐴 (𝑦𝑧 → (abs‘(((𝑧𝐴𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3527, 34syl5bb 272 . . . 4 (𝜑 → (∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3635rexbidv 3200 . . 3 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
3736ralbidv 3135 . 2 (𝜑 → (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑤𝐴 (𝑦𝑤 → (abs‘(((𝑧𝐴𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧𝐴 (𝑦𝑧 → (abs‘(𝐵𝐶)) < 𝑥)))
387, 9, 373bitr2d 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 382  wcel 2145  wral 3061  wrex 3062  wss 3723   class class class wbr 4786  cmpt 4863  wf 6025  cfv 6029  (class class class)co 6791  cc 10134  cr 10135   < clt 10274  cle 10275  cmin 10466  +crp 12028  abscabs 14175  𝑟 crli 14417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-cnex 10192  ax-resscn 10193
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-fv 6037  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-pm 8010  df-rlim 14421
This theorem is referenced by:  rlim2lt  14429  rlim3  14430  rlim0  14440  rlimi  14445  rlimconst  14476  climrlim2  14479  rlimcn1  14520  rlimcn2  14522  chtppilim  25378  pntlem3  25512
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