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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 488 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝐹 ∈ 𝑉) | |
3 | climmpt.2 | . . . 4 ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) | |
4 | fvex 6658 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ∈ V | |
5 | 1, 4 | eqeltri 2886 | . . . . 5 ⊢ 𝑍 ∈ V |
6 | 5 | mptex 6963 | . . . 4 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ∈ V |
7 | 3, 6 | eqeltri 2886 | . . 3 ⊢ 𝐺 ∈ V |
8 | 7 | a1i 11 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝐺 ∈ V) |
9 | simpl 486 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝑀 ∈ ℤ) | |
10 | fveq2 6645 | . . . . 5 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
11 | fvex 6658 | . . . . 5 ⊢ (𝐹‘𝑚) ∈ V | |
12 | 10, 3, 11 | fvmpt 6745 | . . . 4 ⊢ (𝑚 ∈ 𝑍 → (𝐺‘𝑚) = (𝐹‘𝑚)) |
13 | 12 | eqcomd 2804 | . . 3 ⊢ (𝑚 ∈ 𝑍 → (𝐹‘𝑚) = (𝐺‘𝑚)) |
14 | 13 | adantl 485 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = (𝐺‘𝑚)) |
15 | 1, 2, 8, 9, 14 | climeq 14916 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-z 11970 df-uz 12232 df-clim 14837 |
This theorem is referenced by: climmpt2 14922 climrecl 14932 climge0 14933 caurcvg2 15026 caucvg 15027 climfsum 15167 dstfrvclim1 31845 divcnvg 42269 climmptf 42323 |
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