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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 1914 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
| 7 | 5, 6 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) | 
| 8 | climeqf.k | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
| 9 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
| 10 | 8, 9 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) | 
| 11 | climeqf.n | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
| 12 | 11, 9 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) | 
| 13 | 10, 12 | nfeq 2919 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗) | 
| 14 | 7, 13 | nfim 1896 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗)) | 
| 15 | eleq1w 2824 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
| 16 | 15 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) | 
| 17 | fveq2 6906 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
| 18 | fveq2 6906 | . . . . 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 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 2240 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐺‘𝑗)) | 
| 23 | 1, 2, 3, 4, 22 | climeq 15603 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 class class class wbr 5143 ‘cfv 6561 ℤcz 12613 ℤ≥cuz 12878 ⇝ cli 15520 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-neg 11495 df-z 12614 df-uz 12879 df-clim 15524 | 
| This theorem is referenced by: climeqmpt 45712 | 
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