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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 5257 | . 2 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶) | |
3 | nfmpt1 5257 | . 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 7229 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ V) |
8 | climeqmpt.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
9 | 8 | mptexd 7229 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ V) |
10 | climeqmpt.s | . . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴) | |
11 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐴) |
12 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝑍) | |
13 | 11, 12 | sseldd 3984 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐴) |
14 | climeqmpt.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 ∈ 𝑈) | |
15 | eqid 2731 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
16 | 15 | fvmpt2 7010 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) |
17 | 13, 14, 16 | syl2anc 583 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) |
18 | climeqmpt.t | . . . . . . 7 ⊢ (𝜑 → 𝑍 ⊆ 𝐵) | |
19 | 18 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑍 ⊆ 𝐵) |
20 | 19, 12 | sseldd 3984 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝐵) |
21 | eqid 2731 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
22 | 21 | fvmpt2 7010 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐶 ∈ 𝑈) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶) |
23 | 20, 14, 22 | syl2anc 583 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥) = 𝐶) |
24 | 23 | eqcomd 2737 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥)) |
25 | 17, 24 | eqtrd 2771 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ 𝐶)‘𝑥)) |
26 | 1, 2, 3, 4, 5, 7, 9, 25 | climeqf 44704 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝ 𝐷 ↔ (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 ℤcz 12563 ℤ≥cuz 12827 ⇝ cli 15433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-pre-lttri 11187 ax-pre-lttrn 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7415 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-neg 11452 df-z 12564 df-uz 12828 df-clim 15437 |
This theorem is referenced by: smflimsuplem6 45841 smflimsuplem8 45843 |
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