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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climresmpt | Structured version Visualization version GIF version |
Description: A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
climresmpt.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climresmpt.f | ⊢ 𝐹 = (𝑥 ∈ 𝑍 ↦ 𝐴) |
climresmpt.n | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
climresmpt.g | ⊢ 𝐺 = (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) |
Ref | Expression |
---|---|
climresmpt | ⊢ (𝜑 → (𝐺 ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climresmpt.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑍 ↦ 𝐴) | |
2 | 1 | reseq1i 5814 | . . . . 5 ⊢ (𝐹 ↾ (ℤ≥‘𝑁)) = ((𝑥 ∈ 𝑍 ↦ 𝐴) ↾ (ℤ≥‘𝑁)) |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ (ℤ≥‘𝑁)) = ((𝑥 ∈ 𝑍 ↦ 𝐴) ↾ (ℤ≥‘𝑁))) |
4 | climresmpt.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
5 | climresmpt.z | . . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | 4, 5 | eleqtrdi 2900 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
7 | uzss 12253 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
9 | 8, 5 | sseqtrrdi 3966 | . . . . 5 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
10 | resmpt 5872 | . . . . 5 ⊢ ((ℤ≥‘𝑁) ⊆ 𝑍 → ((𝑥 ∈ 𝑍 ↦ 𝐴) ↾ (ℤ≥‘𝑁)) = (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑍 ↦ 𝐴) ↾ (ℤ≥‘𝑁)) = (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴)) |
12 | climresmpt.g | . . . . . 6 ⊢ 𝐺 = (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) | |
13 | 12 | eqcomi 2807 | . . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) = 𝐺 |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) = 𝐺) |
15 | 3, 11, 14 | 3eqtrrd 2838 | . . 3 ⊢ (𝜑 → 𝐺 = (𝐹 ↾ (ℤ≥‘𝑁))) |
16 | 15 | breq1d 5040 | . 2 ⊢ (𝜑 → (𝐺 ⇝ 𝐵 ↔ (𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵)) |
17 | eluzelz 12241 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
18 | 6, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
19 | 5 | fvexi 6659 | . . . . . 6 ⊢ 𝑍 ∈ V |
20 | 19 | mptex 6963 | . . . . 5 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐴) ∈ V |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴) ∈ V) |
22 | 1, 21 | eqeltrid 2894 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
23 | climres 14924 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ V) → ((𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) | |
24 | 18, 22, 23 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) |
25 | 16, 24 | bitrd 282 | 1 ⊢ (𝜑 → (𝐺 ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ↾ cres 5521 ‘cfv 6324 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14833 |
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-rep 5154 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-reu 3113 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-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 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 df-clim 14837 |
This theorem is referenced by: meaiininclem 43125 |
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