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Theorem rlimmptrcl 15640
Description: Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.)
Hypotheses
Ref Expression
rlimabs.1 ((𝜑𝑘𝐴) → 𝐵𝑉)
rlimabs.2 (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)
Assertion
Ref Expression
rlimmptrcl ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Distinct variable groups:   𝐴,𝑘   𝜑,𝑘
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)   𝑉(𝑘)

Proof of Theorem rlimmptrcl
StepHypRef Expression
1 rlimabs.2 . . . 4 (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)
2 rlimf 15533 . . . 4 ((𝑘𝐴𝐵) ⇝𝑟 𝐶 → (𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ)
31, 2syl 17 . . 3 (𝜑 → (𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ)
4 eqid 2734 . . . . 5 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5 rlimabs.1 . . . . 5 ((𝜑𝑘𝐴) → 𝐵𝑉)
64, 5dmmptd 6713 . . . 4 (𝜑 → dom (𝑘𝐴𝐵) = 𝐴)
76feq2d 6722 . . 3 (𝜑 → ((𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ ↔ (𝑘𝐴𝐵):𝐴⟶ℂ))
83, 7mpbid 232 . 2 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
98fvmptelcdm 7132 1 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105   class class class wbr 5147  cmpt 5230  dom cdm 5688  wf 6558  cc 11150  𝑟 crli 15517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-pm 8867  df-rlim 15521
This theorem is referenced by:  rlimabs  15641  rlimcj  15642  rlimre  15643  rlimim  15644  rlimadd  15675  rlimsub  15676  rlimmul  15677  rlimdiv  15678  rlimneg  15679  fsumrlim  15843  dvfsumrlim  26086  rlimcxp  27031  cxploglim2  27036
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