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Mirrors > Home > MPE Home > Th. List > Mathboxes > climeldmeqmpt3 | Structured version Visualization version GIF version |
Description: Two functions that are eventually equal, either both are convergent or both are divergent. TODO: this is more general than climeldmeqmpt 39900 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
climeldmeqmpt3.k | ⊢ Ⅎ𝑘𝜑 |
climeldmeqmpt3.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climeldmeqmpt3.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climeldmeqmpt3.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
climeldmeqmpt3.c | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
climeldmeqmpt3.i | ⊢ (𝜑 → 𝑍 ⊆ 𝐴) |
climeldmeqmpt3.s | ⊢ (𝜑 → 𝑍 ⊆ 𝐶) |
climeldmeqmpt3.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈) |
climeldmeqmpt3.e | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
climeldmeqmpt3 | ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ dom ⇝ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climeldmeqmpt3.z | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climeldmeqmpt3.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | 2 | mptexd 6487 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V) |
4 | climeldmeqmpt3.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
5 | 4 | mptexd 6487 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ V) |
6 | climeldmeqmpt3.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | climeldmeqmpt3.k | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
8 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
9 | 7, 8 | nfan 1828 | . . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
10 | nfcsb1v 3549 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 | |
11 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑘𝑗 | |
12 | 11 | nfcsb1 3548 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐷 |
13 | 10, 12 | nfeq 2776 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷 |
14 | 9, 13 | nfim 1825 | . . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷) |
15 | eleq1 2689 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
16 | 15 | anbi2d 740 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
17 | csbeq1a 3542 | . . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) | |
18 | csbeq1a 3542 | . . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐷 = ⦋𝑗 / 𝑘⦌𝐷) | |
19 | 17, 18 | eqeq12d 2637 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵 = 𝐷 ↔ ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷)) |
20 | 16, 19 | imbi12d 334 | . . . 4 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷))) |
21 | climeldmeqmpt3.e | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷) | |
22 | 14, 20, 21 | chvar 2262 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷) |
23 | climeldmeqmpt3.i | . . . . 5 ⊢ (𝜑 → 𝑍 ⊆ 𝐴) | |
24 | 23 | sselda 3603 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐴) |
25 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑘𝑈 | |
26 | 10, 25 | nfel 2777 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈 |
27 | 9, 26 | nfim 1825 | . . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) |
28 | 17 | eleq1d 2686 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ 𝑈 ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈)) |
29 | 16, 28 | imbi12d 334 | . . . . 5 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈))) |
30 | climeldmeqmpt3.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈) | |
31 | 27, 29, 30 | chvar 2262 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) |
32 | 11 | nfcsb1 3548 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 |
33 | eqid 2622 | . . . . 5 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
34 | 11, 32, 17, 33 | fvmptf 6301 | . . . 4 ⊢ ((𝑗 ∈ 𝐴 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
35 | 24, 31, 34 | syl2anc 693 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
36 | climeldmeqmpt3.s | . . . . 5 ⊢ (𝜑 → 𝑍 ⊆ 𝐶) | |
37 | 36 | sselda 3603 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐶) |
38 | 22, 31 | eqeltrrd 2702 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈) |
39 | eqid 2622 | . . . . 5 ⊢ (𝑘 ∈ 𝐶 ↦ 𝐷) = (𝑘 ∈ 𝐶 ↦ 𝐷) | |
40 | 11, 12, 18, 39 | fvmptf 6301 | . . . 4 ⊢ ((𝑗 ∈ 𝐶 ∧ ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷) |
41 | 37, 38, 40 | syl2anc 693 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷) |
42 | 22, 35, 41 | 3eqtr4d 2666 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗)) |
43 | 1, 3, 5, 6, 42 | climeldmeq 39897 | 1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ dom ⇝ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Vcvv 3200 ⦋csb 3533 ⊆ wss 3574 ↦ cmpt 4729 dom cdm 5114 ‘cfv 5888 ℤcz 11377 ℤ≥cuz 11687 ⇝ cli 14215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 |
This theorem is referenced by: smflimmpt 41016 |
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