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Theorem rlimmptrcl 14679
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 . . . . 5 (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)
2 rlimf 14573 . . . . 5 ((𝑘𝐴𝐵) ⇝𝑟 𝐶 → (𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ)
31, 2syl 17 . . . 4 (𝜑 → (𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ)
4 eqid 2799 . . . . . 6 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5 rlimabs.1 . . . . . 6 ((𝜑𝑘𝐴) → 𝐵𝑉)
64, 5dmmptd 6235 . . . . 5 (𝜑 → dom (𝑘𝐴𝐵) = 𝐴)
76feq2d 6242 . . . 4 (𝜑 → ((𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ ↔ (𝑘𝐴𝐵):𝐴⟶ℂ))
83, 7mpbid 224 . . 3 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
94fmpt 6606 . . 3 (∀𝑘𝐴 𝐵 ∈ ℂ ↔ (𝑘𝐴𝐵):𝐴⟶ℂ)
108, 9sylibr 226 . 2 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
1110r19.21bi 3113 1 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  wral 3089   class class class wbr 4843  cmpt 4922  dom cdm 5312  wf 6097  cc 10222  𝑟 crli 14557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-pm 8098  df-rlim 14561
This theorem is referenced by:  rlimabs  14680  rlimcj  14681  rlimre  14682  rlimim  14683  rlimadd  14714  rlimsub  14715  rlimmul  14716  rlimdiv  14717  rlimneg  14718  fsumrlim  14881  dvfsumrlim  24135  rlimcxp  25052  cxploglim2  25057
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