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Theorem climmptf 45098
Description: Exhibit a function 𝐺 with the same convergence properties as the not-quite-function 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
climmptf.k 𝑘𝐹
climmptf.m (𝜑𝑀 ∈ ℤ)
climmptf.f (𝜑𝐹𝑉)
climmptf.z 𝑍 = (ℤ𝑀)
climmptf.g 𝐺 = (𝑘𝑍 ↦ (𝐹𝑘))
Assertion
Ref Expression
climmptf (𝜑 → (𝐹𝐴𝐺𝐴))
Distinct variable group:   𝑘,𝑍
Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐹(𝑘)   𝐺(𝑘)   𝑀(𝑘)   𝑉(𝑘)

Proof of Theorem climmptf
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 climmptf.m . 2 (𝜑𝑀 ∈ ℤ)
2 climmptf.f . 2 (𝜑𝐹𝑉)
3 climmptf.z . . 3 𝑍 = (ℤ𝑀)
4 climmptf.g . . . 4 𝐺 = (𝑘𝑍 ↦ (𝐹𝑘))
5 nfcv 2899 . . . . 5 𝑗(𝐹𝑘)
6 climmptf.k . . . . . 6 𝑘𝐹
7 nfcv 2899 . . . . . 6 𝑘𝑗
86, 7nffv 6912 . . . . 5 𝑘(𝐹𝑗)
9 fveq2 6902 . . . . 5 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
105, 8, 9cbvmpt 5263 . . . 4 (𝑘𝑍 ↦ (𝐹𝑘)) = (𝑗𝑍 ↦ (𝐹𝑗))
114, 10eqtri 2756 . . 3 𝐺 = (𝑗𝑍 ↦ (𝐹𝑗))
123, 11climmpt 15555 . 2 ((𝑀 ∈ ℤ ∧ 𝐹𝑉) → (𝐹𝐴𝐺𝐴))
131, 2, 12syl2anc 582 1 (𝜑 → (𝐹𝐴𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  wnfc 2879   class class class wbr 5152  cmpt 5235  cfv 6553  cz 12596  cuz 12860  cli 15468
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-pre-lttri 11220  ax-pre-lttrn 11221
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-neg 11485  df-z 12597  df-uz 12861  df-clim 15472
This theorem is referenced by: (None)
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