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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 2923 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)) |
4 | allbutfifvre.2 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | eqid 2821 | . . . 4 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) | |
6 | 4, 5 | allbutfi 41685 | . . 3 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚)) |
7 | 3, 6 | sylib 220 | . 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 765 | . . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝜑) | |
12 | 4 | uztrn2 12263 | . . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍) |
13 | 12 | ssd 41364 | . . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍) |
14 | 13 | sselda 3967 | . . . . . 6 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍) |
15 | 14 | adantll 712 | . . . . 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 6851 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑋 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ) |
18 | 17 | ex 415 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑋 ∈ dom (𝐹‘𝑚) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)) |
19 | 11, 15, 18 | syl2anc 586 | . . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝑋 ∈ dom (𝐹‘𝑚) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)) |
20 | 10, 19 | ralimdaa 3217 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ dom (𝐹‘𝑚) → ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ∈ ℝ)) |
21 | 20 | reximdva 3274 | . 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 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ∪ ciun 4919 ∩ ciin 4920 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 ℝcr 10536 ℤ≥cuz 12244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-neg 10873 df-z 11983 df-uz 12245 |
This theorem is referenced by: fnlimabslt 41980 |
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