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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 45623 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 7243 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ∈ V) |
4 | climeldmeqmpt3.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
5 | 4 | mptexd 7243 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ V) |
6 | climeldmeqmpt3.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | climeldmeqmpt3.k | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
8 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
9 | 7, 8 | nfan 1896 | . . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
10 | nfcsb1v 3932 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 | |
11 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑘𝑗 | |
12 | 11 | nfcsb1 3931 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐷 |
13 | 10, 12 | nfeq 2916 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷 |
14 | 9, 13 | nfim 1893 | . . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷) |
15 | eleq1w 2821 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
16 | 15 | anbi2d 630 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
17 | csbeq1a 3921 | . . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) | |
18 | csbeq1a 3921 | . . . . . 6 ⊢ (𝑘 = 𝑗 → 𝐷 = ⦋𝑗 / 𝑘⦌𝐷) | |
19 | 17, 18 | eqeq12d 2750 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵 = 𝐷 ↔ ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷)) |
20 | 16, 19 | imbi12d 344 | . . . 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 | chvarfv 2237 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐷) |
23 | climeldmeqmpt3.i | . . . . 5 ⊢ (𝜑 → 𝑍 ⊆ 𝐴) | |
24 | 23 | sselda 3994 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐴) |
25 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑘𝑈 | |
26 | 10, 25 | nfel 2917 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈 |
27 | 9, 26 | nfim 1893 | . . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) |
28 | 17 | eleq1d 2823 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ 𝑈 ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈)) |
29 | 16, 28 | imbi12d 344 | . . . . 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 | chvarfv 2237 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) |
32 | 11 | nfcsb1 3931 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 |
33 | eqid 2734 | . . . . 5 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
34 | 11, 32, 17, 33 | fvmptf 7036 | . . . 4 ⊢ ((𝑗 ∈ 𝐴 ∧ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑈) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
35 | 24, 31, 34 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐵) |
36 | climeldmeqmpt3.s | . . . . 5 ⊢ (𝜑 → 𝑍 ⊆ 𝐶) | |
37 | 36 | sselda 3994 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝐶) |
38 | 22, 31 | eqeltrrd 2839 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈) |
39 | eqid 2734 | . . . . 5 ⊢ (𝑘 ∈ 𝐶 ↦ 𝐷) = (𝑘 ∈ 𝐶 ↦ 𝐷) | |
40 | 11, 12, 18, 39 | fvmptf 7036 | . . . 4 ⊢ ((𝑗 ∈ 𝐶 ∧ ⦋𝑗 / 𝑘⦌𝐷 ∈ 𝑈) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷) |
41 | 37, 38, 40 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐷) |
42 | 22, 35, 41 | 3eqtr4d 2784 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑗) = ((𝑘 ∈ 𝐶 ↦ 𝐷)‘𝑗)) |
43 | 1, 3, 5, 6, 42 | climeldmeq 45620 | 1 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ dom ⇝ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 Vcvv 3477 ⦋csb 3907 ⊆ wss 3962 ↦ cmpt 5230 dom cdm 5688 ‘cfv 6562 ℤcz 12610 ℤ≥cuz 12875 ⇝ cli 15516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 |
This theorem is referenced by: smflimmpt 46765 |
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