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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 1139 | . . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐹:𝐵⟶ℝ*) | |
2 | reex 10785 | . . . . . . 7 ⊢ ℝ ∈ V | |
3 | 2 | ssex 5199 | . . . . . 6 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V) |
4 | 3 | 3ad2ant1 1135 | . . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐵 ∈ V) |
5 | xrex 12548 | . . . . . 6 ⊢ ℝ* ∈ V | |
6 | 5 | a1i 11 | . . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → ℝ* ∈ V) |
7 | fex2 7689 | . . . . 5 ⊢ ((𝐹:𝐵⟶ℝ* ∧ 𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V) | |
8 | 1, 4, 6, 7 | syl3anc 1373 | . . . 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 15000 | . . . 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 5051 | . 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ 𝐴 ≤ inf(ran 𝐺, ℝ*, < ))) |
13 | 9 | limsupgf 15001 | . . . . 5 ⊢ 𝐺:ℝ⟶ℝ* |
14 | frn 6530 | . . . . 5 ⊢ (𝐺:ℝ⟶ℝ* → ran 𝐺 ⊆ ℝ*) | |
15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ ran 𝐺 ⊆ ℝ* |
16 | simp3 1140 | . . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐴 ∈ ℝ*) | |
17 | infxrgelb 12890 | . . . 4 ⊢ ((ran 𝐺 ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥)) | |
18 | 15, 16, 17 | sylancr 590 | . . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑥 ∈ ran 𝐺 𝐴 ≤ 𝑥)) |
19 | ffn 6523 | . . . . 5 ⊢ (𝐺:ℝ⟶ℝ* → 𝐺 Fn ℝ) | |
20 | 13, 19 | ax-mp 5 | . . . 4 ⊢ 𝐺 Fn ℝ |
21 | breq2 5043 | . . . . 5 ⊢ (𝑥 = (𝐺‘𝑗) → (𝐴 ≤ 𝑥 ↔ 𝐴 ≤ (𝐺‘𝑗))) | |
22 | 21 | ralrn 6885 | . . . 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 | bitrdi 290 | . 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ inf(ran 𝐺, ℝ*, < ) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗))) |
25 | 12, 24 | bitrd 282 | 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 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 ∩ cin 3852 ⊆ wss 3853 class class class wbr 5039 ↦ cmpt 5120 ran crn 5537 “ cima 5539 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 supcsup 9034 infcinf 9035 ℝcr 10693 +∞cpnf 10829 ℝ*cxr 10831 < clt 10832 ≤ cle 10833 [,)cico 12902 lim supclsp 14996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-limsup 14997 |
This theorem is referenced by: limsuplt 15005 limsupbnd1 15008 limsupbnd2 15009 mbflimsup 24517 limsupge 42920 |
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