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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 5151 | . . . . . . 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 3176 | . . . . 5 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∀𝑗 ∈ 𝐵 (𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))) |
8 | nfv 1916 | . . . . . . 7 ⊢ Ⅎ𝑙(𝑖 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) | |
9 | nfv 1916 | . . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑙 | |
10 | limsupbnd1f.1 | . . . . . . . . . 10 ⊢ Ⅎ𝑗𝐹 | |
11 | nfcv 2902 | . . . . . . . . . 10 ⊢ Ⅎ𝑗𝑙 | |
12 | 10, 11 | nffv 6901 | . . . . . . . . 9 ⊢ Ⅎ𝑗(𝐹‘𝑙) |
13 | nfcv 2902 | . . . . . . . . 9 ⊢ Ⅎ𝑗 ≤ | |
14 | nfcv 2902 | . . . . . . . . 9 ⊢ Ⅎ𝑗𝐴 | |
15 | 12, 13, 14 | nfbr 5195 | . . . . . . . 8 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝐴 |
16 | 9, 15 | nfim 1898 | . . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴) |
17 | breq2 5152 | . . . . . . . 8 ⊢ (𝑗 = 𝑙 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑙)) | |
18 | fveq2 6891 | . . . . . . . . 9 ⊢ (𝑗 = 𝑙 → (𝐹‘𝑗) = (𝐹‘𝑙)) | |
19 | 18 | breq1d 5158 | . . . . . . . 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 3302 | . . . . . 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 3234 | . . 3 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴) ↔ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)) |
25 | 4, 24 | sylib 217 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐵 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝐴)) |
26 | 1, 2, 3, 25 | limsupbnd1 15433 | 1 ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 Ⅎwnfc 2882 ∀wral 3060 ∃wrex 3069 ⊆ wss 3948 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 ℝcr 11115 ℝ*cxr 11254 ≤ cle 11256 lim supclsp 15421 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-ico 13337 df-limsup 15422 |
This theorem is referenced by: limsuppnflem 44885 |
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