Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupequz | Structured version Visualization version GIF version |
Description: Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupequz.1 | ⊢ Ⅎ𝑘𝜑 |
limsupequz.2 | ⊢ Ⅎ𝑘𝐹 |
limsupequz.3 | ⊢ Ⅎ𝑘𝐺 |
limsupequz.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
limsupequz.5 | ⊢ (𝜑 → 𝐹 Fn (ℤ≥‘𝑀)) |
limsupequz.6 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
limsupequz.7 | ⊢ (𝜑 → 𝐺 Fn (ℤ≥‘𝑁)) |
limsupequz.8 | ⊢ (𝜑 → 𝐾 ∈ ℤ) |
limsupequz.9 | ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘)) |
Ref | Expression |
---|---|
limsupequz | ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . 2 ⊢ Ⅎ𝑗𝜑 | |
2 | limsupequz.4 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | limsupequz.5 | . 2 ⊢ (𝜑 → 𝐹 Fn (ℤ≥‘𝑀)) | |
4 | limsupequz.6 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
5 | limsupequz.7 | . 2 ⊢ (𝜑 → 𝐺 Fn (ℤ≥‘𝑁)) | |
6 | limsupequz.8 | . 2 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
7 | limsupequz.1 | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
8 | nfv 1843 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ (ℤ≥‘𝐾) | |
9 | 7, 8 | nfan 1828 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) |
10 | limsupequz.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
11 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
12 | 10, 11 | nffv 6198 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
13 | limsupequz.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
14 | 13, 11 | nffv 6198 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
15 | 12, 14 | nfeq 2776 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗) |
16 | 9, 15 | nfim 1825 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
17 | eleq1 2689 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (ℤ≥‘𝐾) ↔ 𝑗 ∈ (ℤ≥‘𝐾))) | |
18 | 17 | anbi2d 740 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ↔ (𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)))) |
19 | fveq2 6191 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
20 | fveq2 6191 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
21 | 19, 20 | eqeq12d 2637 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑗) = (𝐺‘𝑗))) |
22 | 18, 21 | imbi12d 334 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘)) ↔ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑗) = (𝐺‘𝑗)))) |
23 | limsupequz.9 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘)) | |
24 | 16, 22, 23 | chvar 2262 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
25 | 1, 2, 3, 4, 5, 6, 24 | limsupequzlem 39954 | 1 ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 Fn wfn 5883 ‘cfv 5888 ℤcz 11377 ℤ≥cuz 11687 lim supclsp 14201 |
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-int 4476 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-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 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-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-ico 12181 df-limsup 14202 |
This theorem is referenced by: limsupequzmptlem 39960 smflimsuplem2 41027 |
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