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

Theorem rlimmptrcl 15496
Description: Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.)
Hypotheses
Ref Expression
rlimabs.1 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)
rlimabs.2 (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ 𝐡) β‡π‘Ÿ 𝐢)
Assertion
Ref Expression
rlimmptrcl ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)
Distinct variable groups:   𝐴,π‘˜   πœ‘,π‘˜
Allowed substitution hints:   𝐡(π‘˜)   𝐢(π‘˜)   𝑉(π‘˜)

Proof of Theorem rlimmptrcl
StepHypRef Expression
1 rlimabs.2 . . . 4 (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ 𝐡) β‡π‘Ÿ 𝐢)
2 rlimf 15389 . . . 4 ((π‘˜ ∈ 𝐴 ↦ 𝐡) β‡π‘Ÿ 𝐢 β†’ (π‘˜ ∈ 𝐴 ↦ 𝐡):dom (π‘˜ ∈ 𝐴 ↦ 𝐡)βŸΆβ„‚)
31, 2syl 17 . . 3 (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ 𝐡):dom (π‘˜ ∈ 𝐴 ↦ 𝐡)βŸΆβ„‚)
4 eqid 2733 . . . . 5 (π‘˜ ∈ 𝐴 ↦ 𝐡) = (π‘˜ ∈ 𝐴 ↦ 𝐡)
5 rlimabs.1 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)
64, 5dmmptd 6647 . . . 4 (πœ‘ β†’ dom (π‘˜ ∈ 𝐴 ↦ 𝐡) = 𝐴)
76feq2d 6655 . . 3 (πœ‘ β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡):dom (π‘˜ ∈ 𝐴 ↦ 𝐡)βŸΆβ„‚ ↔ (π‘˜ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚))
83, 7mpbid 231 . 2 (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚)
98fvmptelcdm 7062 1 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∈ wcel 2107   class class class wbr 5106   ↦ cmpt 5189  dom cdm 5634  βŸΆwf 6493  β„‚cc 11054   β‡π‘Ÿ crli 15373
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-pm 8771  df-rlim 15377
This theorem is referenced by:  rlimabs  15497  rlimcj  15498  rlimre  15499  rlimim  15500  rlimadd  15531  rlimaddOLD  15532  rlimsub  15533  rlimmul  15534  rlimmulOLD  15535  rlimdiv  15536  rlimneg  15537  fsumrlim  15701  dvfsumrlim  25411  rlimcxp  26339  cxploglim2  26344
  Copyright terms: Public domain W3C validator