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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 5128 | . 2 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶) | |
3 | nfmpt1 5128 | . 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 6964 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V) |
8 | climeqmpt.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
9 | 8 | mptexd 6964 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ V) |
10 | climeqmpt.s | . . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴) | |
11 | 10 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐴) |
12 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝑍) | |
13 | 11, 12 | sseldd 3916 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐴) |
14 | climeqmpt.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 ∈ 𝑈) | |
15 | eqid 2798 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
16 | 15 | fvmpt2 6756 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) |
17 | 13, 14, 16 | syl2anc 587 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) |
18 | climeqmpt.t | . . . . . . 7 ⊢ (𝜑 → 𝑍 ⊆ 𝐵) | |
19 | 18 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐵) |
20 | 19, 12 | sseldd 3916 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐵) |
21 | eqid 2798 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
22 | 21 | fvmpt2 6756 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶) |
23 | 20, 14, 22 | syl2anc 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶) |
24 | 23 | eqcomd 2804 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥)) |
25 | 17, 24 | eqtrd 2833 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥)) |
26 | 1, 2, 3, 4, 5, 7, 9, 25 | climeqf 42330 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝ 𝐷 ↔ (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-z 11970 df-uz 12232 df-clim 14837 |
This theorem is referenced by: smflimsuplem6 43456 smflimsuplem8 43458 |
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