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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 1912 | . 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 1912 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ (ℤ≥‘𝐾) | |
9 | 7, 8 | nfan 1897 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) |
10 | limsupequz.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
11 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
12 | 10, 11 | nffv 6917 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
13 | limsupequz.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
14 | 13, 11 | nffv 6917 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
15 | 12, 14 | nfeq 2917 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) = (𝐺‘𝑗) |
16 | 9, 15 | nfim 1894 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑗) = (𝐺‘𝑗)) |
17 | eleq1w 2822 | . . . . 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 6907 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
20 | fveq2 6907 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
21 | 19, 20 | eqeq12d 2751 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = (𝐺‘𝑘) ↔ (𝐹‘𝑗) = (𝐺‘𝑗))) |
22 | 18, 21 | imbi12d 344 | . . 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 45678 | 1 ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 Fn wfn 6558 ‘cfv 6563 ℤcz 12611 ℤ≥cuz 12876 lim supclsp 15503 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-ico 13390 df-limsup 15504 |
This theorem is referenced by: limsupequzmptlem 45684 smflimsuplem2 46777 |
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