Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 | syl6eleq 2698 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
4 | allbutfifvre.2 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | eqid 2610 | . . . 4 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) | |
6 | 4, 5 | allbutfi 38557 | . . 3 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚)) |
7 | 3, 6 | sylib 207 | . 2 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚)) |
8 | allbutfifvre.1 | . . . . 5 ⊢ Ⅎ𝑚𝜑 | |
9 | nfv 1830 | . . . . 5 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍 | |
10 | 8, 9 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍) |
11 | simpll 786 | . . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑) | |
12 | 4 | uztrn2 11581 | . . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍) |
13 | 12 | ssd 38278 | . . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍) |
14 | 13 | sselda 3568 | . . . . . 6 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍) |
15 | 14 | adantll 746 | . . . . 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 6267 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑋 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ) |
18 | 17 | ex 449 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑋 ∈ dom (𝐹‘𝑚) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)) |
19 | 11, 15, 18 | syl2anc 691 | . . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝑋 ∈ dom (𝐹‘𝑚) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)) |
20 | 10, 19 | ralimdaa 2941 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚) → ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ)) |
21 | 20 | reximdva 3000 | . 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 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∪ ciun 4455 ∩ ciin 4456 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 ℝcr 9814 ℤ≥cuz 11563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-neg 10148 df-z 11255 df-uz 11564 |
This theorem is referenced by: fnlimabslt 38746 |
Copyright terms: Public domain | W3C validator |