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Theorem rlimmptrcl 15593
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 15486 . . . 4 ((𝑘𝐴𝐵) ⇝𝑟 𝐶 → (𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ)
31, 2syl 17 . . 3 (𝜑 → (𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ)
4 eqid 2725 . . . . 5 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5 rlimabs.1 . . . . 5 ((𝜑𝑘𝐴) → 𝐵𝑉)
64, 5dmmptd 6701 . . . 4 (𝜑 → dom (𝑘𝐴𝐵) = 𝐴)
76feq2d 6709 . . 3 (𝜑 → ((𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ ↔ (𝑘𝐴𝐵):𝐴⟶ℂ))
83, 7mpbid 231 . 2 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
98fvmptelcdm 7122 1 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098   class class class wbr 5149  cmpt 5232  dom cdm 5678  wf 6545  cc 11143  𝑟 crli 15470
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-pm 8848  df-rlim 15474
This theorem is referenced by:  rlimabs  15594  rlimcj  15595  rlimre  15596  rlimim  15597  rlimadd  15628  rlimaddOLD  15629  rlimsub  15630  rlimmul  15631  rlimmulOLD  15632  rlimdiv  15633  rlimneg  15634  fsumrlim  15798  dvfsumrlim  26015  rlimcxp  26956  cxploglim2  26961
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