![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > climmptf | Structured version Visualization version GIF version |
Description: Exhibit a function 𝐺 with the same convergence properties as the not-quite-function 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
climmptf.k | ⊢ Ⅎ𝑘𝐹 |
climmptf.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climmptf.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
climmptf.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climmptf.g | ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) |
Ref | Expression |
---|---|
climmptf | ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climmptf.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | climmptf.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
3 | climmptf.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | climmptf.g | . . . 4 ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) | |
5 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑗(𝐹‘𝑘) | |
6 | climmptf.k | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
7 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
8 | 6, 7 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
9 | fveq2 6906 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
10 | 5, 8, 9 | cbvmpt 5258 | . . . 4 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) |
11 | 4, 10 | eqtri 2762 | . . 3 ⊢ 𝐺 = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) |
12 | 3, 11 | climmpt 15603 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
13 | 1, 2, 12 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 Ⅎwnfc 2887 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: (None) |
Copyright terms: Public domain | W3C validator |