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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupbnd1f | Structured version Visualization version GIF version |
Description: If a sequence is eventually at most 𝐴, then the limsup is also at most 𝐴. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupbnd1f.1 | ⊢ Ⅎ𝑗𝐹 |
limsupbnd1f.2 | ⊢ (𝜑 → 𝐵 ⊆ ℝ) |
limsupbnd1f.3 | ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) |
limsupbnd1f.4 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
limsupbnd1f.5 | ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)) |
Ref | Expression |
---|---|
limsupbnd1f | ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupbnd1f.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ ℝ) | |
2 | limsupbnd1f.3 | . 2 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) | |
3 | limsupbnd1f.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | limsupbnd1f.5 | . . 3 ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)) | |
5 | breq1 5169 | . . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗)) | |
6 | 5 | imbi1d 341 | . . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))) |
7 | 6 | ralbidv 3184 | . . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑗 ∈ 𝐵 (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))) |
8 | nfv 1913 | . . . . . . 7 ⊢ Ⅎ𝑙(𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) | |
9 | nfv 1913 | . . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑙 | |
10 | limsupbnd1f.1 | . . . . . . . . . 10 ⊢ Ⅎ𝑗𝐹 | |
11 | nfcv 2908 | . . . . . . . . . 10 ⊢ Ⅎ𝑗𝑙 | |
12 | 10, 11 | nffv 6932 | . . . . . . . . 9 ⊢ Ⅎ𝑗(𝐹‘𝑙) |
13 | nfcv 2908 | . . . . . . . . 9 ⊢ Ⅎ𝑗 ≤ | |
14 | nfcv 2908 | . . . . . . . . 9 ⊢ Ⅎ𝑗𝐴 | |
15 | 12, 13, 14 | nfbr 5213 | . . . . . . . 8 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝐴 |
16 | 9, 15 | nfim 1895 | . . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴) |
17 | breq2 5170 | . . . . . . . 8 ⊢ (𝑗 = 𝑙 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑙)) | |
18 | fveq2 6922 | . . . . . . . . 9 ⊢ (𝑗 = 𝑙 → (𝐹‘𝑗) = (𝐹‘𝑙)) | |
19 | 18 | breq1d 5176 | . . . . . . . 8 ⊢ (𝑗 = 𝑙 → ((𝐹‘𝑗) ≤ 𝐴 ↔ (𝐹‘𝑙) ≤ 𝐴)) |
20 | 17, 19 | imbi12d 344 | . . . . . . 7 ⊢ (𝑗 = 𝑙 → ((𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴))) |
21 | 8, 16, 20 | cbvralw 3312 | . . . . . 6 ⊢ (∀𝑗 ∈ 𝐵 (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)) |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐵 (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴))) |
23 | 7, 22 | bitrd 279 | . . . 4 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴))) |
24 | 23 | cbvrexvw 3244 | . . 3 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)) |
25 | 4, 24 | sylib 218 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)) |
26 | 1, 2, 3, 25 | limsupbnd1 15530 | 1 ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2108 Ⅎwnfc 2893 ∀wral 3067 ∃wrex 3076 ⊆ wss 3976 class class class wbr 5166 ⟶wf 6571 ‘cfv 6575 ℝcr 11185 ℝ*cxr 11325 ≤ cle 11327 lim supclsp 15518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-sup 9513 df-inf 9514 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-ico 13415 df-limsup 15519 |
This theorem is referenced by: limsuppnflem 45633 |
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