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 2979 | . . . . 5 ⊢ Ⅎ𝑗(𝐹‘𝑘) | |
6 | climmptf.k | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
7 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
8 | 6, 7 | nffv 6682 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
9 | fveq2 6672 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
10 | 5, 8, 9 | cbvmpt 5169 | . . . 4 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) |
11 | 4, 10 | eqtri 2846 | . . 3 ⊢ 𝐺 = (𝑗 ∈ 𝑍 ↦ (𝐹‘𝑗)) |
12 | 3, 11 | climmpt 14930 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
13 | 1, 2, 12 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2963 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 ℤcz 11984 ℤ≥cuz 12246 ⇝ cli 14843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-neg 10875 df-z 11985 df-uz 12247 df-clim 14847 |
This theorem is referenced by: (None) |
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