Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 1915 | . 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 1915 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ (ℤ≥‘𝐾) | |
9 | 7, 8 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) |
10 | limsupequz.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
11 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
12 | 10, 11 | nffv 6682 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
13 | limsupequz.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
14 | 13, 11 | nffv 6682 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
15 | 12, 14 | nfeq 2993 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗) |
16 | 9, 15 | nfim 1897 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
17 | eleq1w 2897 | . . . . 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 6672 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
20 | fveq2 6672 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
21 | 19, 20 | eqeq12d 2839 | . . . 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 2242 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
25 | 1, 2, 3, 4, 5, 6, 24 | limsupequzlem 42010 | 1 ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 Fn wfn 6352 ‘cfv 6357 ℤcz 11984 ℤ≥cuz 12246 lim supclsp 14829 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-ico 12747 df-limsup 14830 |
This theorem is referenced by: limsupequzmptlem 42016 smflimsuplem2 43102 |
Copyright terms: Public domain | W3C validator |