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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 4965 | . . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗)) | |
6 | 5 | imbi1d 343 | . . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))) |
7 | 6 | ralbidv 3164 | . . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑗 ∈ 𝐵 (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))) |
8 | nfv 1892 | . . . . . . 7 ⊢ Ⅎ𝑙(𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) | |
9 | nfv 1892 | . . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑙 | |
10 | limsupbnd1f.1 | . . . . . . . . . 10 ⊢ Ⅎ𝑗𝐹 | |
11 | nfcv 2949 | . . . . . . . . . 10 ⊢ Ⅎ𝑗𝑙 | |
12 | 10, 11 | nffv 6548 | . . . . . . . . 9 ⊢ Ⅎ𝑗(𝐹‘𝑙) |
13 | nfcv 2949 | . . . . . . . . 9 ⊢ Ⅎ𝑗 ≤ | |
14 | nfcv 2949 | . . . . . . . . 9 ⊢ Ⅎ𝑗𝐴 | |
15 | 12, 13, 14 | nfbr 5009 | . . . . . . . 8 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝐴 |
16 | 9, 15 | nfim 1878 | . . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴) |
17 | breq2 4966 | . . . . . . . 8 ⊢ (𝑗 = 𝑙 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑙)) | |
18 | fveq2 6538 | . . . . . . . . 9 ⊢ (𝑗 = 𝑙 → (𝐹‘𝑗) = (𝐹‘𝑙)) | |
19 | 18 | breq1d 4972 | . . . . . . . 8 ⊢ (𝑗 = 𝑙 → ((𝐹‘𝑗) ≤ 𝐴 ↔ (𝐹‘𝑙) ≤ 𝐴)) |
20 | 17, 19 | imbi12d 346 | . . . . . . 7 ⊢ (𝑗 = 𝑙 → ((𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴))) |
21 | 8, 16, 20 | cbvral 3399 | . . . . . 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 280 | . . . 4 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴))) |
24 | 23 | cbvrexv 3404 | . . 3 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)) |
25 | 4, 24 | sylib 219 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)) |
26 | 1, 2, 3, 25 | limsupbnd1 14673 | 1 ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1522 ∈ wcel 2081 Ⅎwnfc 2933 ∀wral 3105 ∃wrex 3106 ⊆ wss 3859 class class class wbr 4962 ⟶wf 6221 ‘cfv 6225 ℝcr 10382 ℝ*cxr 10520 ≤ cle 10522 lim supclsp 14661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-po 5362 df-so 5363 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-ico 12594 df-limsup 14662 |
This theorem is referenced by: limsuppnflem 41533 |
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