Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupubuzmpt | Structured version Visualization version GIF version |
Description: If the limsup is not +∞, then the function is eventually bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupubuzmpt.j | ⊢ Ⅎ𝑗𝜑 |
limsupubuzmpt.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
limsupubuzmpt.b | ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ) |
limsupubuzmpt.n | ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵)) ≠ +∞) |
Ref | Expression |
---|---|
limsupubuzmpt | ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfmpt1 5156 | . . . 4 ⊢ Ⅎ𝑗(𝑗 ∈ 𝑍 ↦ 𝐵) | |
2 | limsupubuzmpt.z | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | limsupubuzmpt.j | . . . . 5 ⊢ Ⅎ𝑗𝜑 | |
4 | limsupubuzmpt.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ) | |
5 | eqid 2821 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 ↦ 𝐵) = (𝑗 ∈ 𝑍 ↦ 𝐵) | |
6 | 3, 4, 5 | fmptdf 6875 | . . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ) |
7 | limsupubuzmpt.n | . . . 4 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵)) ≠ +∞) | |
8 | 1, 2, 6, 7 | limsupubuz 41987 | . . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 ((𝑗 ∈ 𝑍 ↦ 𝐵)‘𝑗) ≤ 𝑦) |
9 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ 𝐵) = (𝑗 ∈ 𝑍 ↦ 𝐵)) |
10 | 9, 4 | fvmpt2d 6775 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑗 ∈ 𝑍 ↦ 𝐵)‘𝑗) = 𝐵) |
11 | 10 | breq1d 5068 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑗 ∈ 𝑍 ↦ 𝐵)‘𝑗) ≤ 𝑦 ↔ 𝐵 ≤ 𝑦)) |
12 | 3, 11 | ralbida 3230 | . . . 4 ⊢ (𝜑 → (∀𝑗 ∈ 𝑍 ((𝑗 ∈ 𝑍 ↦ 𝐵)‘𝑗) ≤ 𝑦 ↔ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑦)) |
13 | 12 | rexbidv 3297 | . . 3 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 ((𝑗 ∈ 𝑍 ↦ 𝐵)‘𝑗) ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑦)) |
14 | 8, 13 | mpbid 234 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑦) |
15 | breq2 5062 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝐵 ≤ 𝑦 ↔ 𝐵 ≤ 𝑥)) | |
16 | 15 | ralbidv 3197 | . . 3 ⊢ (𝑦 = 𝑥 → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑦 ↔ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥)) |
17 | 16 | cbvrexvw 3450 | . 2 ⊢ (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥) |
18 | 14, 17 | sylib 220 | 1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 class class class wbr 5058 ↦ cmpt 5138 ‘cfv 6349 ℝcr 10530 +∞cpnf 10666 ≤ cle 10670 ℤ≥cuz 12237 lim supclsp 14821 |
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-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
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-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-ico 12738 df-fz 12887 df-fl 13156 df-ceil 13157 df-limsup 14822 |
This theorem is referenced by: smflimsuplem2 43089 smflimsuplem5 43092 |
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