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Theorem allbutfifvre 41946
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, 2eleqtrdi 2921 . . 3 (𝜑𝑋 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
4 allbutfifvre.2 . . . 4 𝑍 = (ℤ𝑀)
5 eqid 2819 . . . 4 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
64, 5allbutfi 41655 . . 3 (𝑋 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚))
73, 6sylib 220 . 2 (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚))
8 allbutfifvre.1 . . . . 5 𝑚𝜑
9 nfv 1909 . . . . 5 𝑚 𝑛𝑍
108, 9nfan 1894 . . . 4 𝑚(𝜑𝑛𝑍)
11 simpll 765 . . . . 5 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
124uztrn2 12254 . . . . . . . 8 ((𝑛𝑍𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
1312ssd 41335 . . . . . . 7 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
1413sselda 3965 . . . . . 6 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1514adantll 712 . . . . 5 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
16 allbutfifvre.3 . . . . . . 7 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
1716ffvelrnda 6844 . . . . . 6 (((𝜑𝑚𝑍) ∧ 𝑋 ∈ dom (𝐹𝑚)) → ((𝐹𝑚)‘𝑋) ∈ ℝ)
1817ex 415 . . . . 5 ((𝜑𝑚𝑍) → (𝑋 ∈ dom (𝐹𝑚) → ((𝐹𝑚)‘𝑋) ∈ ℝ))
1911, 15, 18syl2anc 586 . . . 4 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑋 ∈ dom (𝐹𝑚) → ((𝐹𝑚)‘𝑋) ∈ ℝ))
2010, 19ralimdaa 3215 . . 3 ((𝜑𝑛𝑍) → (∀𝑚 ∈ (ℤ𝑛)𝑋 ∈ dom (𝐹𝑚) → ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ∈ ℝ))
2120reximdva 3272 . 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 398   = wceq 1531  wnf 1778  wcel 2108  wral 3136  wrex 3137   ciun 4910   ciin 4911  dom cdm 5548  wf 6344  cfv 6348  cr 10528  cuz 12235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-pre-lttri 10603  ax-pre-lttrn 10604
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-neg 10865  df-z 11974  df-uz 12236
This theorem is referenced by:  fnlimabslt  41950
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