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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 1912 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
7 | 5, 6 | nfan 1897 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
8 | climeqf.k | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
9 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
10 | 8, 9 | nffv 6917 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
11 | climeqf.n | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
12 | 11, 9 | nffv 6917 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
13 | 10, 12 | nfeq 2917 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗) |
14 | 7, 13 | nfim 1894 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
15 | eleq1w 2822 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
16 | 15 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
17 | fveq2 6907 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
18 | fveq2 6907 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
19 | 17, 18 | eqeq12d 2751 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑗) = (𝐺‘𝑗))) |
20 | 16, 19 | imbi12d 344 | . . 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 15600 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 class class class wbr 5148 ‘cfv 6563 ℤcz 12611 ℤ≥cuz 12876 ⇝ cli 15517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-neg 11493 df-z 12612 df-uz 12877 df-clim 15521 |
This theorem is referenced by: climeqmpt 45653 |
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