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Mirrors > Home > MPE Home > Th. List > Mathboxes > allbutfifvre | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
allbutfifvre.1 | ⊢ Ⅎ𝑚𝜑 |
allbutfifvre.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
allbutfifvre.3 | ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) |
allbutfifvre.4 | ⊢ 𝐷 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) |
allbutfifvre.5 | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
allbutfifvre | ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | allbutfifvre.5 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
2 | allbutfifvre.4 | . . . 4 ⊢ 𝐷 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) | |
3 | 1, 2 | eleqtrdi 2900 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
4 | allbutfifvre.2 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | eqid 2798 | . . . 4 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) | |
6 | 4, 5 | allbutfi 42029 | . . 3 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚)) |
7 | 3, 6 | sylib 221 | . 2 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚)) |
8 | allbutfifvre.1 | . . . . 5 ⊢ Ⅎ𝑚𝜑 | |
9 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍 | |
10 | 8, 9 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍) |
11 | simpll 766 | . . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑) | |
12 | 4 | uztrn2 12250 | . . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍) |
13 | 12 | ssd 41716 | . . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍) |
14 | 13 | sselda 3915 | . . . . . 6 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍) |
15 | 14 | adantll 713 | . . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍) |
16 | allbutfifvre.3 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) | |
17 | 16 | ffvelrnda 6828 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑋 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ) |
18 | 17 | ex 416 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑋 ∈ dom (𝐹‘𝑚) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)) |
19 | 11, 15, 18 | syl2anc 587 | . . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝑋 ∈ dom (𝐹‘𝑚) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)) |
20 | 10, 19 | ralimdaa 3181 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚) → ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ)) |
21 | 20 | reximdva 3233 | . 2 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ)) |
22 | 7, 21 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ∪ ciun 4881 ∩ ciin 4882 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 ℝcr 10525 ℤ≥cuz 12231 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
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 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-z 11970 df-uz 12232 |
This theorem is referenced by: fnlimabslt 42321 |
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