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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 476 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝐹 ∈ 𝑉) | |
3 | climmpt.2 | . . . 4 ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) | |
4 | fvex 6113 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ∈ V | |
5 | 1, 4 | eqeltri 2684 | . . . . 5 ⊢ 𝑍 ∈ V |
6 | 5 | mptex 6390 | . . . 4 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ∈ V |
7 | 3, 6 | eqeltri 2684 | . . 3 ⊢ 𝐺 ∈ V |
8 | 7 | a1i 11 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝐺 ∈ V) |
9 | simpl 472 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → 𝑀 ∈ ℤ) | |
10 | fveq2 6103 | . . . . 5 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
11 | fvex 6113 | . . . . 5 ⊢ (𝐹‘𝑚) ∈ V | |
12 | 10, 3, 11 | fvmpt 6191 | . . . 4 ⊢ (𝑚 ∈ 𝑍 → (𝐺‘𝑚) = (𝐹‘𝑚)) |
13 | 12 | eqcomd 2616 | . . 3 ⊢ (𝑚 ∈ 𝑍 → (𝐹‘𝑚) = (𝐺‘𝑚)) |
14 | 13 | adantl 481 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) = (𝐺‘𝑚)) |
15 | 1, 2, 8, 9, 14 | climeq 14146 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 ℤcz 11254 ℤ≥cuz 11563 ⇝ cli 14063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-neg 10148 df-z 11255 df-uz 11564 df-clim 14067 |
This theorem is referenced by: climmpt2 14152 climrecl 14162 climge0 14163 caurcvg2 14256 caucvg 14257 climfsum 14393 dstfrvclim1 29866 divcnvg 38694 |
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