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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 5993 | . . . . 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 2851 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 7 | uzss 12901 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
| 8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
| 9 | 8, 5 | sseqtrrdi 4025 | . . . . 5 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) |
| 10 | resmpt 6055 | . . . . 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 2746 | . . . . 5 ⊢ (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) = 𝐺 |
| 14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (ℤ≥‘𝑁) ↦ 𝐴) = 𝐺) |
| 15 | 3, 11, 14 | 3eqtrrd 2782 | . . 3 ⊢ (𝜑 → 𝐺 = (𝐹 ↾ (ℤ≥‘𝑁))) |
| 16 | 15 | breq1d 5153 | . 2 ⊢ (𝜑 → (𝐺 ⇝ 𝐵 ↔ (𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵)) |
| 17 | eluzelz 12888 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
| 18 | 6, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 19 | 5 | fvexi 6920 | . . . . . 6 ⊢ 𝑍 ∈ V |
| 20 | 19 | mptex 7243 | . . . . 5 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐴) ∈ V |
| 21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴) ∈ V) |
| 22 | 1, 21 | eqeltrid 2845 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
| 23 | climres 15611 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐹 ∈ V) → ((𝐹 ↾ (ℤ≥‘𝑁)) ⇝ 𝐵 ↔ 𝐹 ⇝ 𝐵)) | |
| 24 | 18, 22, 23 | syl2anc 584 | . 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 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ↾ cres 5687 ‘cfv 6561 ℤcz 12613 ℤ≥cuz 12878 ⇝ cli 15520 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 df-clim 15524 |
| This theorem is referenced by: meaiininclem 46501 |
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