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Theorem allbutfifvre 39343
Description: Given a sequence of real-valued functions, and 𝑋 that belongs to all but finitely many domains, then its function value is ultimately a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
allbutfifvre.1 𝑚𝜑
allbutfifvre.2 𝑍 = (ℤ𝑀)
allbutfifvre.3 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
allbutfifvre.4 𝐷 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
allbutfifvre.5 (𝜑𝑋𝐷)
Assertion
Ref Expression
allbutfifvre (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ∈ ℝ)
Distinct variable groups:   𝑚,𝑋,𝑛   𝑚,𝑍   𝜑,𝑛
Allowed substitution hints:   𝜑(𝑚)   𝐷(𝑚,𝑛)   𝐹(𝑚,𝑛)   𝑀(𝑚,𝑛)   𝑍(𝑛)

Proof of Theorem allbutfifvre
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 allbutfifvre.5 . . . 4 (𝜑𝑋𝐷)
2 allbutfifvre.4 . . . 4 𝐷 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
31, 2syl6eleq 2708 . . 3 (𝜑𝑋 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
4 allbutfifvre.2 . . . 4 𝑍 = (ℤ𝑀)
5 eqid 2621 . . . 4 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
64, 5allbutfi 39115 . . 3 (𝑋 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚))
73, 6sylib 208 . 2 (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚))
8 allbutfifvre.1 . . . . 5 𝑚𝜑
9 nfv 1840 . . . . 5 𝑚 𝑛𝑍
108, 9nfan 1825 . . . 4 𝑚(𝜑𝑛𝑍)
11 simpll 789 . . . . 5 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
124uztrn2 11665 . . . . . . . 8 ((𝑛𝑍𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
1312ssd 38774 . . . . . . 7 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
1413sselda 3588 . . . . . 6 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1514adantll 749 . . . . 5 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
16 allbutfifvre.3 . . . . . . 7 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
1716ffvelrnda 6325 . . . . . 6 (((𝜑𝑚𝑍) ∧ 𝑋 ∈ dom (𝐹𝑚)) → ((𝐹𝑚)‘𝑋) ∈ ℝ)
1817ex 450 . . . . 5 ((𝜑𝑚𝑍) → (𝑋 ∈ dom (𝐹𝑚) → ((𝐹𝑚)‘𝑋) ∈ ℝ))
1911, 15, 18syl2anc 692 . . . 4 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑋 ∈ dom (𝐹𝑚) → ((𝐹𝑚)‘𝑋) ∈ ℝ))
2010, 19ralimdaa 2954 . . 3 ((𝜑𝑛𝑍) → (∀𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚) → ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ∈ ℝ))
2120reximdva 3013 . 2 (𝜑 → (∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚) → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ∈ ℝ))
227, 21mpd 15 1 (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wnf 1705  wcel 1987  wral 2908  wrex 2909   ciun 4492   ciin 4493  dom cdm 5084  wf 5853  cfv 5857  cr 9895  cuz 11647
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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-pre-lttri 9970  ax-pre-lttrn 9971
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  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-nel 2894  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-neg 10229  df-z 11338  df-uz 11648
This theorem is referenced by:  fnlimabslt  39347
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