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Mathbox for Glauco Siliprandi |
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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 2899 | . . . . 5 ⊢ Ⅎ𝑗(𝐹‘𝑘) | |
6 | climmptf.k | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
7 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
8 | 6, 7 | nffv 6912 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
9 | fveq2 6902 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
10 | 5, 8, 9 | cbvmpt 5263 | . . . 4 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) |
11 | 4, 10 | eqtri 2756 | . . 3 ⊢ 𝐺 = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) |
12 | 3, 11 | climmpt 15555 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
13 | 1, 2, 12 | syl2anc 582 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2879 class class class wbr 5152 ↦ cmpt 5235 ‘cfv 6553 ℤcz 12596 ℤ≥cuz 12860 ⇝ cli 15468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-neg 11485 df-z 12597 df-uz 12861 df-clim 15472 |
This theorem is referenced by: (None) |
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