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Theorem rlimmptrcl 14272
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 14166 . . . . 5 ((𝑘𝐴𝐵) ⇝𝑟 𝐶 → (𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ)
31, 2syl 17 . . . 4 (𝜑 → (𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ)
4 eqid 2621 . . . . . 6 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5 rlimabs.1 . . . . . 6 ((𝜑𝑘𝐴) → 𝐵𝑉)
64, 5dmmptd 5981 . . . . 5 (𝜑 → dom (𝑘𝐴𝐵) = 𝐴)
76feq2d 5988 . . . 4 (𝜑 → ((𝑘𝐴𝐵):dom (𝑘𝐴𝐵)⟶ℂ ↔ (𝑘𝐴𝐵):𝐴⟶ℂ))
83, 7mpbid 222 . . 3 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
94fmpt 6337 . . 3 (∀𝑘𝐴 𝐵 ∈ ℂ ↔ (𝑘𝐴𝐵):𝐴⟶ℂ)
108, 9sylibr 224 . 2 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
1110r19.21bi 2927 1 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2907   class class class wbr 4613  cmpt 4673  dom cdm 5074  wf 5843  cc 9878  𝑟 crli 14150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-pm 7805  df-rlim 14154
This theorem is referenced by:  rlimabs  14273  rlimcj  14274  rlimre  14275  rlimim  14276  rlimadd  14307  rlimsub  14308  rlimmul  14309  rlimdiv  14310  rlimneg  14311  fsumrlim  14470  dvfsumrlim  23698  rlimcxp  24600  cxploglim2  24605
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