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 5851 | . . . . 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 2925 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
7 | uzss 12268 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
9 | 8, 5 | sseqtrrdi 4020 | . . . . 5 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
10 | resmpt 5907 | . . . . 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 2832 | . . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) = 𝐺 |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) = 𝐺) |
15 | 3, 11, 14 | 3eqtrrd 2863 | . . 3 ⊢ (𝜑 → 𝐺 = (𝐹 ↾ (ℤ≥‘𝑁))) |
16 | 15 | breq1d 5078 | . 2 ⊢ (𝜑 → (𝐺 ⇝ 𝐵 ↔ (𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵)) |
17 | eluzelz 12256 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
18 | 6, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
19 | 5 | fvexi 6686 | . . . . . 6 ⊢ 𝑍 ∈ V |
20 | 19 | mptex 6988 | . . . . 5 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐴) ∈ V |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴) ∈ V) |
22 | 1, 21 | eqeltrid 2919 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
23 | climres 14934 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ V) → ((𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) | |
24 | 18, 22, 23 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) |
25 | 16, 24 | bitrd 281 | 1 ⊢ (𝜑 → (𝐺 ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 ↾ cres 5559 ‘cfv 6357 ℤcz 11984 ℤ≥cuz 12246 ⇝ cli 14843 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-neg 10875 df-z 11985 df-uz 12247 df-clim 14847 |
This theorem is referenced by: meaiininclem 42775 |
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