Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupmnf | Structured version Visualization version GIF version |
Description: The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to an upper interval of real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupmnf.j | ⊢ Ⅎ𝑗𝐹 |
limsupmnf.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
limsupmnf.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
Ref | Expression |
---|---|
limsupmnf | ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupmnf.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | limsupmnf.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
3 | eqid 2622 | . . 3 ⊢ (𝑖 ∈ ℝ ↦ sup((𝐹 “ (𝑖[,)+∞)), ℝ*, < )) = (𝑖 ∈ ℝ ↦ sup((𝐹 “ (𝑖[,)+∞)), ℝ*, < )) | |
4 | 1, 2, 3 | limsupmnflem 39952 | . 2 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦))) |
5 | breq2 4657 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥)) | |
6 | 5 | imbi2d 330 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥))) |
7 | 6 | ralbidv 2986 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥))) |
8 | 7 | rexbidv 3052 | . . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥))) |
9 | breq1 4656 | . . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑖 ≤ 𝑙 ↔ 𝑘 ≤ 𝑙)) | |
10 | 9 | imbi1d 331 | . . . . . . . . 9 ⊢ (𝑖 = 𝑘 → ((𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥))) |
11 | 10 | ralbidv 2986 | . . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥))) |
12 | nfv 1843 | . . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑘 ≤ 𝑙 | |
13 | limsupmnf.j | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐹 | |
14 | nfcv 2764 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑙 | |
15 | 13, 14 | nffv 6198 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙) |
16 | nfcv 2764 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗 ≤ | |
17 | nfcv 2764 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑥 | |
18 | 15, 16, 17 | nfbr 4699 | . . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥 |
19 | 12, 18 | nfim 1825 | . . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) |
20 | nfv 1843 | . . . . . . . . . 10 ⊢ Ⅎ𝑙(𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) | |
21 | breq2 4657 | . . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → (𝑘 ≤ 𝑙 ↔ 𝑘 ≤ 𝑗)) | |
22 | fveq2 6191 | . . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗)) | |
23 | 22 | breq1d 4663 | . . . . . . . . . . 11 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)) |
24 | 21, 23 | imbi12d 334 | . . . . . . . . . 10 ⊢ (𝑙 = 𝑗 → ((𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
25 | 19, 20, 24 | cbvral 3167 | . . . . . . . . 9 ⊢ (∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
26 | 25 | a1i 11 | . . . . . . . 8 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑘 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
27 | 11, 26 | bitrd 268 | . . . . . . 7 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
28 | 27 | cbvrexv 3172 | . . . . . 6 ⊢ (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
29 | 28 | a1i 11 | . . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
30 | 8, 29 | bitrd 268 | . . . 4 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
31 | 30 | cbvralv 3171 | . . 3 ⊢ (∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
32 | 31 | a1i 11 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑖 ∈ ℝ ∀𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 → (𝐹‘𝑙) ≤ 𝑦) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
33 | 4, 32 | bitrd 268 | 1 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 Ⅎwnfc 2751 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supcsup 8346 ℝcr 9935 +∞cpnf 10071 -∞cmnf 10072 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 [,)cico 12177 lim supclsp 14201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-ico 12181 df-limsup 14202 |
This theorem is referenced by: limsupre2lem 39956 limsupmnfuzlem 39958 |
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