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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 2897 | . . . . 5 ⊢ Ⅎ𝑗(𝐹‘𝑘) | |
6 | climmptf.k | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
7 | nfcv 2897 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
8 | 6, 7 | nffv 6894 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
9 | fveq2 6884 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
10 | 5, 8, 9 | cbvmpt 5252 | . . . 4 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) |
11 | 4, 10 | eqtri 2754 | . . 3 ⊢ 𝐺 = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) |
12 | 3, 11 | climmpt 15518 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
13 | 1, 2, 12 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2877 class class class wbr 5141 ↦ cmpt 5224 ‘cfv 6536 ℤcz 12559 ℤ≥cuz 12823 ⇝ cli 15431 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-neg 11448 df-z 12560 df-uz 12824 df-clim 15435 |
This theorem is referenced by: (None) |
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