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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 5638 | . . . . 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 | syl6eleq 2868 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
7 | uzss 12013 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
9 | 8, 5 | syl6sseqr 3870 | . . . . 5 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
10 | resmpt 5699 | . . . . 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 2786 | . . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) = 𝐺 |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) = 𝐺) |
15 | 3, 11, 14 | 3eqtrrd 2818 | . . 3 ⊢ (𝜑 → 𝐺 = (𝐹 ↾ (ℤ≥‘𝑁))) |
16 | 15 | breq1d 4896 | . 2 ⊢ (𝜑 → (𝐺 ⇝ 𝐵 ↔ (𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵)) |
17 | eluzelz 12002 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
18 | 6, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
19 | 5 | fvexi 6460 | . . . . . 6 ⊢ 𝑍 ∈ V |
20 | 19 | mptex 6758 | . . . . 5 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐴) ∈ V |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴) ∈ V) |
22 | 1, 21 | syl5eqel 2862 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
23 | climres 14714 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ V) → ((𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) | |
24 | 18, 22, 23 | syl2anc 579 | . 2 ⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) |
25 | 16, 24 | bitrd 271 | 1 ⊢ (𝜑 → (𝐺 ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ⊆ wss 3791 class class class wbr 4886 ↦ cmpt 4965 ↾ cres 5357 ‘cfv 6135 ℤcz 11728 ℤ≥cuz 11992 ⇝ cli 14623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-neg 10609 df-z 11729 df-uz 11993 df-clim 14627 |
This theorem is referenced by: meaiininclem 41620 |
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