![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 5937 | . . . . 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 2844 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
7 | uzss 12794 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
9 | 8, 5 | sseqtrrdi 3999 | . . . . 5 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
10 | resmpt 5995 | . . . . 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 2742 | . . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) = 𝐺 |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) = 𝐺) |
15 | 3, 11, 14 | 3eqtrrd 2778 | . . 3 ⊢ (𝜑 → 𝐺 = (𝐹 ↾ (ℤ≥‘𝑁))) |
16 | 15 | breq1d 5119 | . 2 ⊢ (𝜑 → (𝐺 ⇝ 𝐵 ↔ (𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵)) |
17 | eluzelz 12781 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
18 | 6, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
19 | 5 | fvexi 6860 | . . . . . 6 ⊢ 𝑍 ∈ V |
20 | 19 | mptex 7177 | . . . . 5 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐴) ∈ V |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴) ∈ V) |
22 | 1, 21 | eqeltrid 2838 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
23 | climres 15466 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ V) → ((𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) | |
24 | 18, 22, 23 | syl2anc 585 | . 2 ⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) |
25 | 16, 24 | bitrd 279 | 1 ⊢ (𝜑 → (𝐺 ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ⊆ wss 3914 class class class wbr 5109 ↦ cmpt 5192 ↾ cres 5639 ‘cfv 6500 ℤcz 12507 ℤ≥cuz 12771 ⇝ cli 15375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-neg 11396 df-z 12508 df-uz 12772 df-clim 15379 |
This theorem is referenced by: meaiininclem 44817 |
Copyright terms: Public domain | W3C validator |