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Mirrors > Home > MPE Home > Th. List > climmpt | Structured version Visualization version GIF version |
Description: Exhibit a function 𝐺 with the same convergence properties as the not-quite-function 𝐹. (Contributed by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
2clim.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climmpt.2 | ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) |
Ref | Expression |
---|---|
climmpt | ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2clim.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | simpr 484 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝐹 ∈ 𝑉) | |
3 | climmpt.2 | . . . 4 ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) | |
4 | fvex 6919 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ∈ V | |
5 | 1, 4 | eqeltri 2834 | . . . . 5 ⊢ 𝑍 ∈ V |
6 | 5 | mptex 7242 | . . . 4 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ∈ V |
7 | 3, 6 | eqeltri 2834 | . . 3 ⊢ 𝐺 ∈ V |
8 | 7 | a1i 11 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝐺 ∈ V) |
9 | simpl 482 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝑀 ∈ ℤ) | |
10 | fveq2 6906 | . . . . 5 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
11 | fvex 6919 | . . . . 5 ⊢ (𝐹‘𝑚) ∈ V | |
12 | 10, 3, 11 | fvmpt 7015 | . . . 4 ⊢ (𝑚 ∈ 𝑍 → (𝐺‘𝑚) = (𝐹‘𝑚)) |
13 | 12 | eqcomd 2740 | . . 3 ⊢ (𝑚 ∈ 𝑍 → (𝐹‘𝑚) = (𝐺‘𝑚)) |
14 | 13 | adantl 481 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = (𝐺‘𝑚)) |
15 | 1, 2, 8, 9, 14 | climeq 15599 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 ℤcz 12610 ℤ≥cuz 12875 ⇝ cli 15516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-pre-lttri 11226 ax-pre-lttrn 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-neg 11492 df-z 12611 df-uz 12876 df-clim 15520 |
This theorem is referenced by: climmpt2 15605 climrecl 15615 climge0 15616 caurcvg2 15710 caucvg 15711 climfsum 15852 dstfrvclim1 34458 divcnvg 45582 climmptf 45636 |
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