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 1911 | . 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 1911 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ (ℤ≥‘𝐾) | |
9 | 7, 8 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) |
10 | limsupequz.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
11 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
12 | 10, 11 | nffv 6674 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
13 | limsupequz.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
14 | 13, 11 | nffv 6674 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
15 | 12, 14 | nfeq 2991 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗) |
16 | 9, 15 | nfim 1893 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
17 | eleq1w 2895 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (ℤ≥‘𝐾) ↔ 𝑗 ∈ (ℤ≥‘𝐾))) | |
18 | 17 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ↔ (𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)))) |
19 | fveq2 6664 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
20 | fveq2 6664 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
21 | 19, 20 | eqeq12d 2837 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑗) = (𝐺‘𝑗))) |
22 | 18, 21 | imbi12d 347 | . . 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 | chvarfv 2238 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
25 | 1, 2, 3, 4, 5, 6, 24 | limsupequzlem 41996 | 1 ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 Ⅎwnfc 2961 Fn wfn 6344 ‘cfv 6349 ℤcz 11975 ℤ≥cuz 12237 lim supclsp 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-ico 12738 df-limsup 14822 |
This theorem is referenced by: limsupequzmptlem 42002 smflimsuplem2 43089 |
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