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Mirrors > Home > MPE Home > Th. List > limsuple | Structured version Visualization version GIF version |
Description: The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.) |
Ref | Expression |
---|---|
limsupval.1 | ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
Ref | Expression |
---|---|
limsuple | ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1133 | . . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐹:𝐵⟶ℝ*) | |
2 | reex 10627 | . . . . . . 7 ⊢ ℝ ∈ V | |
3 | 2 | ssex 5224 | . . . . . 6 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V) |
4 | 3 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐵 ∈ V) |
5 | xrex 12385 | . . . . . 6 ⊢ ℝ* ∈ V | |
6 | 5 | a1i 11 | . . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → ℝ* ∈ V) |
7 | fex2 7637 | . . . . 5 ⊢ ((𝐹:𝐵⟶ℝ* ∧ 𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V) | |
8 | 1, 4, 6, 7 | syl3anc 1367 | . . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐹 ∈ V) |
9 | limsupval.1 | . . . . 5 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
10 | 9 | limsupval 14830 | . . . 4 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < )) |
11 | 8, 10 | syl 17 | . . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < )) |
12 | 11 | breq2d 5077 | . 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ 𝐴 ≤ inf(ran 𝐺, ℝ*, < ))) |
13 | 9 | limsupgf 14831 | . . . . 5 ⊢ 𝐺:ℝ⟶ℝ* |
14 | frn 6519 | . . . . 5 ⊢ (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*) | |
15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ ran 𝐺 ⊆ ℝ* |
16 | simp3 1134 | . . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐴 ∈ ℝ*) | |
17 | infxrgelb 12727 | . . . 4 ⊢ ((ran 𝐺 ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥)) | |
18 | 15, 16, 17 | sylancr 589 | . . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥)) |
19 | ffn 6513 | . . . . 5 ⊢ (𝐺:ℝ⟶ℝ* → 𝐺 Fn ℝ) | |
20 | 13, 19 | ax-mp 5 | . . . 4 ⊢ 𝐺 Fn ℝ |
21 | breq2 5069 | . . . . 5 ⊢ (𝑥 = (𝐺‘𝑗) → (𝐴 ≤ 𝑥 ↔ 𝐴 ≤ (𝐺‘𝑗))) | |
22 | 21 | ralrn 6853 | . . . 4 ⊢ (𝐺 Fn ℝ → (∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥 ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗))) |
23 | 20, 22 | ax-mp 5 | . . 3 ⊢ (∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥 ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗)) |
24 | 18, 23 | syl6bb 289 | . 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗))) |
25 | 12, 24 | bitrd 281 | 1 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 class class class wbr 5065 ↦ cmpt 5145 ran crn 5555 “ cima 5557 Fn wfn 6349 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 supcsup 8903 infcinf 8904 ℝcr 10535 +∞cpnf 10671 ℝ*cxr 10673 < clt 10674 ≤ cle 10675 [,)cico 12739 lim supclsp 14826 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-limsup 14827 |
This theorem is referenced by: limsuplt 14835 limsupbnd1 14838 limsupbnd2 14839 mbflimsup 24266 limsupge 42040 |
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