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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climeqmpt | 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 |
---|---|
climeqmpt.x | ⊢ Ⅎ𝑥𝜑 |
climeqmpt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
climeqmpt.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
climeqmpt.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climeqmpt.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climeqmpt.s | ⊢ (𝜑 → 𝑍 ⊆ 𝐴) |
climeqmpt.t | ⊢ (𝜑 → 𝑍 ⊆ 𝐵) |
climeqmpt.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 ∈ 𝑈) |
Ref | Expression |
---|---|
climeqmpt | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝ 𝐷 ↔ (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climeqmpt.x | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfmpt1 5255 | . 2 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶) | |
3 | nfmpt1 5255 | . 2 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐵 ↦ 𝐶) | |
4 | climeqmpt.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | climeqmpt.z | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | climeqmpt.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | 6 | mptexd 7221 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V) |
8 | climeqmpt.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
9 | 8 | mptexd 7221 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ V) |
10 | climeqmpt.s | . . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴) | |
11 | 10 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐴) |
12 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝑍) | |
13 | 11, 12 | sseldd 3982 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐴) |
14 | climeqmpt.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 ∈ 𝑈) | |
15 | eqid 2733 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
16 | 15 | fvmpt2 7005 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) |
17 | 13, 14, 16 | syl2anc 585 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) |
18 | climeqmpt.t | . . . . . . 7 ⊢ (𝜑 → 𝑍 ⊆ 𝐵) | |
19 | 18 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐵) |
20 | 19, 12 | sseldd 3982 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐵) |
21 | eqid 2733 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
22 | 21 | fvmpt2 7005 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶) |
23 | 20, 14, 22 | syl2anc 585 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶) |
24 | 23 | eqcomd 2739 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥)) |
25 | 17, 24 | eqtrd 2773 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥)) |
26 | 1, 2, 3, 4, 5, 7, 9, 25 | climeqf 44339 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝ 𝐷 ↔ (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 ℤcz 12554 ℤ≥cuz 12818 ⇝ cli 15424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-neg 11443 df-z 12555 df-uz 12819 df-clim 15428 |
This theorem is referenced by: smflimsuplem6 45476 smflimsuplem8 45478 |
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