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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 5155 | . 2 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶) | |
3 | nfmpt1 5155 | . 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 6978 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V) |
8 | climeqmpt.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
9 | 8 | mptexd 6978 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ V) |
10 | climeqmpt.s | . . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴) | |
11 | 10 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐴) |
12 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝑍) | |
13 | 11, 12 | sseldd 3965 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐴) |
14 | climeqmpt.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 ∈ 𝑈) | |
15 | eqid 2818 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
16 | 15 | fvmpt2 6771 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) |
17 | 13, 14, 16 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) |
18 | climeqmpt.t | . . . . . . 7 ⊢ (𝜑 → 𝑍 ⊆ 𝐵) | |
19 | 18 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐵) |
20 | 19, 12 | sseldd 3965 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐵) |
21 | eqid 2818 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
22 | 21 | fvmpt2 6771 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶) |
23 | 20, 14, 22 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶) |
24 | 23 | eqcomd 2824 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥)) |
25 | 17, 24 | eqtrd 2853 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥)) |
26 | 1, 2, 3, 4, 5, 7, 9, 25 | climeqf 41845 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝ 𝐷 ↔ (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 ‘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-rep 5181 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-reu 3142 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-iun 4912 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: smflimsuplem6 42976 smflimsuplem8 42978 |
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