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 1922 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
7 | 5, 6 | nfan 1907 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
8 | climeqf.k | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
9 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
10 | 8, 9 | nffv 6727 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
11 | climeqf.n | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
12 | 11, 9 | nffv 6727 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
13 | 10, 12 | nfeq 2917 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗) |
14 | 7, 13 | nfim 1904 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
15 | eleq1w 2820 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
16 | 15 | anbi2d 632 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
17 | fveq2 6717 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
18 | fveq2 6717 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
19 | 17, 18 | eqeq12d 2753 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑗) = (𝐺‘𝑗))) |
20 | 16, 19 | imbi12d 348 | . . 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 2238 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
23 | 1, 2, 3, 4, 22 | climeq 15128 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2884 class class class wbr 5053 ‘cfv 6380 ℤcz 12176 ℤ≥cuz 12438 ⇝ cli 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-pre-lttri 10803 ax-pre-lttrn 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-neg 11065 df-z 12177 df-uz 12439 df-clim 15049 |
This theorem is referenced by: climeqmpt 42913 |
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