Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > limsuppnfd | Structured version Visualization version GIF version |
Description: If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsuppnfd.j | ⊢ Ⅎ𝑗𝐹 |
limsuppnfd.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
limsuppnfd.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
limsuppnfd.u | ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) |
Ref | Expression |
---|---|
limsuppnfd | ⊢ (𝜑 → (lim sup‘𝐹) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsuppnfd.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | limsuppnfd.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
3 | limsuppnfd.u | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) | |
4 | breq1 5071 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ (𝐹‘𝑗) ↔ 𝑦 ≤ (𝐹‘𝑗))) | |
5 | 4 | anbi2d 630 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) |
6 | 5 | rexbidv 3299 | . . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) |
7 | breq1 5071 | . . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑗 ↔ 𝑖 ≤ 𝑗)) | |
8 | 7 | anbi1d 631 | . . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ (𝑖 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) |
9 | 8 | rexbidv 3299 | . . . . 5 ⊢ (𝑘 = 𝑖 → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ ∃𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)))) |
10 | nfv 1915 | . . . . . . 7 ⊢ Ⅎ𝑙(𝑖 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) | |
11 | nfv 1915 | . . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ≤ 𝑙 | |
12 | nfcv 2979 | . . . . . . . . 9 ⊢ Ⅎ𝑗𝑦 | |
13 | nfcv 2979 | . . . . . . . . 9 ⊢ Ⅎ𝑗 ≤ | |
14 | limsuppnfd.j | . . . . . . . . . 10 ⊢ Ⅎ𝑗𝐹 | |
15 | nfcv 2979 | . . . . . . . . . 10 ⊢ Ⅎ𝑗𝑙 | |
16 | 14, 15 | nffv 6682 | . . . . . . . . 9 ⊢ Ⅎ𝑗(𝐹‘𝑙) |
17 | 12, 13, 16 | nfbr 5115 | . . . . . . . 8 ⊢ Ⅎ𝑗 𝑦 ≤ (𝐹‘𝑙) |
18 | 11, 17 | nfan 1900 | . . . . . . 7 ⊢ Ⅎ𝑗(𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)) |
19 | breq2 5072 | . . . . . . . 8 ⊢ (𝑗 = 𝑙 → (𝑖 ≤ 𝑗 ↔ 𝑖 ≤ 𝑙)) | |
20 | fveq2 6672 | . . . . . . . . 9 ⊢ (𝑗 = 𝑙 → (𝐹‘𝑗) = (𝐹‘𝑙)) | |
21 | 20 | breq2d 5080 | . . . . . . . 8 ⊢ (𝑗 = 𝑙 → (𝑦 ≤ (𝐹‘𝑗) ↔ 𝑦 ≤ (𝐹‘𝑙))) |
22 | 19, 21 | anbi12d 632 | . . . . . . 7 ⊢ (𝑗 = 𝑙 → ((𝑖 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)))) |
23 | 10, 18, 22 | cbvrexw 3444 | . . . . . 6 ⊢ (∃𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))) |
24 | 23 | a1i 11 | . . . . 5 ⊢ (𝑘 = 𝑖 → (∃𝑗 ∈ 𝐴 (𝑖 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)))) |
25 | 9, 24 | bitrd 281 | . . . 4 ⊢ (𝑘 = 𝑖 → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑦 ≤ (𝐹‘𝑗)) ↔ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙)))) |
26 | 6, 25 | cbvral2vw 3463 | . . 3 ⊢ (∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))) |
27 | 3, 26 | sylib 220 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ ∀𝑖 ∈ ℝ ∃𝑙 ∈ 𝐴 (𝑖 ≤ 𝑙 ∧ 𝑦 ≤ (𝐹‘𝑙))) |
28 | eqid 2823 | . 2 ⊢ (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
29 | 1, 2, 27, 28 | limsuppnfdlem 41989 | 1 ⊢ (𝜑 → (lim sup‘𝐹) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 Ⅎwnfc 2963 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 “ cima 5560 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 supcsup 8906 ℝcr 10538 +∞cpnf 10674 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 [,)cico 12743 lim supclsp 14829 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-ico 12747 df-limsup 14830 |
This theorem is referenced by: limsupub 41992 limsuppnflem 41998 |
Copyright terms: Public domain | W3C validator |