Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climeqf | Structured version Visualization version GIF version |
Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
climeqf.p | ⊢ Ⅎ𝑘𝜑 |
climeqf.k | ⊢ Ⅎ𝑘𝐹 |
climeqf.n | ⊢ Ⅎ𝑘𝐺 |
climeqf.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climeqf.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climeqf.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
climeqf.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
climeqf.e | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
Ref | Expression |
---|---|
climeqf | ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climeqf.z | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climeqf.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
3 | climeqf.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
4 | climeqf.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | climeqf.p | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
6 | nfv 1906 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
7 | 5, 6 | nfan 1891 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
8 | climeqf.k | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
9 | nfcv 2974 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
10 | 8, 9 | nffv 6673 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
11 | climeqf.n | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
12 | 11, 9 | nffv 6673 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
13 | 10, 12 | nfeq 2988 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗) |
14 | 7, 13 | nfim 1888 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
15 | eleq1w 2892 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
16 | 15 | anbi2d 628 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
17 | fveq2 6663 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
18 | fveq2 6663 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
19 | 17, 18 | eqeq12d 2834 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑗) = (𝐺‘𝑗))) |
20 | 16, 19 | imbi12d 346 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗)))) |
21 | climeqf.e | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
22 | 14, 20, 21 | chvarfv 2232 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
23 | 1, 2, 3, 4, 22 | climeq 14912 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 class class class wbr 5057 ‘cfv 6348 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-neg 10861 df-z 11970 df-uz 12232 df-clim 14833 |
This theorem is referenced by: climeqmpt 41854 |
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