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Theorem climmptf 42189
 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 2982 . . . . 5 𝑗(𝐹𝑘)
6 climmptf.k . . . . . 6 𝑘𝐹
7 nfcv 2982 . . . . . 6 𝑘𝑗
86, 7nffv 6669 . . . . 5 𝑘(𝐹𝑗)
9 fveq2 6659 . . . . 5 (𝑘 = 𝑗 → (𝐹𝑘) = (𝐹𝑗))
105, 8, 9cbvmpt 5154 . . . 4 (𝑘𝑍 ↦ (𝐹𝑘)) = (𝑗𝑍 ↦ (𝐹𝑗))
114, 10eqtri 2847 . . 3 𝐺 = (𝑗𝑍 ↦ (𝐹𝑗))
123, 11climmpt 14926 . 2 ((𝑀 ∈ ℤ ∧ 𝐹𝑉) → (𝐹𝐴𝐺𝐴))
131, 2, 12syl2anc 587 1 (𝜑 → (𝐹𝐴𝐺𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2115  Ⅎwnfc 2962   class class class wbr 5053   ↦ cmpt 5133  ‘cfv 6344  ℤcz 11976  ℤ≥cuz 12238   ⇝ cli 14839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-cnex 10587  ax-resscn 10588  ax-pre-lttri 10605  ax-pre-lttrn 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-po 5462  df-so 5463  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-neg 10867  df-z 11977  df-uz 12239  df-clim 14843 This theorem is referenced by: (None)
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