MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rlimmptrcl Structured version   Visualization version   GIF version

Theorem rlimmptrcl 14958
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 14852 . . . 4 ((𝑘𝐴𝐵) ⇝𝑟 𝐶 → (𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ)
31, 2syl 17 . . 3 (𝜑 → (𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ)
4 eqid 2821 . . . . 5 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5 rlimabs.1 . . . . 5 ((𝜑𝑘𝐴) → 𝐵𝑉)
64, 5dmmptd 6488 . . . 4 (𝜑 → dom (𝑘𝐴𝐵) = 𝐴)
76feq2d 6495 . . 3 (𝜑 → ((𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ ↔ (𝑘𝐴𝐵):𝐴⟶ℂ))
83, 7mpbid 234 . 2 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
98fvmptelrn 6872 1 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2110   class class class wbr 5059  cmpt 5139  dom cdm 5550  wf 6346  cc 10529  𝑟 crli 14836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-pm 8403  df-rlim 14840
This theorem is referenced by:  rlimabs  14959  rlimcj  14960  rlimre  14961  rlimim  14962  rlimadd  14993  rlimsub  14994  rlimmul  14995  rlimdiv  14996  rlimneg  14997  fsumrlim  15160  dvfsumrlim  24622  rlimcxp  25545  cxploglim2  25550
  Copyright terms: Public domain W3C validator