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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsuplt2 | Structured version Visualization version GIF version |
Description: The defining property of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsuplt2.1 | ⊢ (𝜑 → 𝐵 ⊆ ℝ) |
limsuplt2.2 | ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) |
limsuplt2.3 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
Ref | Expression |
---|---|
limsuplt2 | ⊢ (𝜑 → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑘 ∈ ℝ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsuplt2.1 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ) | |
2 | limsuplt2.2 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) | |
3 | limsuplt2.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | eqid 2797 | . . . 4 ⊢ (𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
5 | 4 | limsuplt 14674 | . . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑖 ∈ ℝ ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) < 𝐴)) |
6 | 1, 2, 3, 5 | syl3anc 1364 | . 2 ⊢ (𝜑 → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑖 ∈ ℝ ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) < 𝐴)) |
7 | oveq1 7030 | . . . . . . . 8 ⊢ (𝑗 = 𝑖 → (𝑗[,)+∞) = (𝑖[,)+∞)) | |
8 | 7 | imaeq2d 5813 | . . . . . . 7 ⊢ (𝑗 = 𝑖 → (𝐹 “ (𝑗[,)+∞)) = (𝐹 “ (𝑖[,)+∞))) |
9 | 8 | ineq1d 4114 | . . . . . 6 ⊢ (𝑗 = 𝑖 → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*)) |
10 | 9 | supeq1d 8763 | . . . . 5 ⊢ (𝑗 = 𝑖 → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) |
11 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → 𝑖 ∈ ℝ) | |
12 | xrltso 12388 | . . . . . . 7 ⊢ < Or ℝ* | |
13 | 12 | supex 8780 | . . . . . 6 ⊢ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) ∈ V) |
15 | 4, 10, 11, 14 | fvmptd3 6664 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < )) |
16 | 15 | breq1d 4978 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → (((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) < 𝐴 ↔ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴)) |
17 | 16 | rexbidva 3261 | . 2 ⊢ (𝜑 → (∃𝑖 ∈ ℝ ((𝑗 ∈ ℝ ↦ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑖) < 𝐴 ↔ ∃𝑖 ∈ ℝ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴)) |
18 | oveq1 7030 | . . . . . . . 8 ⊢ (𝑖 = 𝑘 → (𝑖[,)+∞) = (𝑘[,)+∞)) | |
19 | 18 | imaeq2d 5813 | . . . . . . 7 ⊢ (𝑖 = 𝑘 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑘[,)+∞))) |
20 | 19 | ineq1d 4114 | . . . . . 6 ⊢ (𝑖 = 𝑘 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
21 | 20 | supeq1d 8763 | . . . . 5 ⊢ (𝑖 = 𝑘 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) = sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
22 | 21 | breq1d 4978 | . . . 4 ⊢ (𝑖 = 𝑘 → (sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴 ↔ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴)) |
23 | 22 | cbvrexv 3406 | . . 3 ⊢ (∃𝑖 ∈ ℝ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴 ↔ ∃𝑘 ∈ ℝ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴) |
24 | 23 | a1i 11 | . 2 ⊢ (𝜑 → (∃𝑖 ∈ ℝ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴 ↔ ∃𝑘 ∈ ℝ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴)) |
25 | 6, 17, 24 | 3bitrd 306 | 1 ⊢ (𝜑 → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑘 ∈ ℝ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∃wrex 3108 Vcvv 3440 ∩ cin 3864 ⊆ wss 3865 class class class wbr 4968 ↦ cmpt 5047 “ cima 5453 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 supcsup 8757 ℝcr 10389 +∞cpnf 10525 ℝ*cxr 10527 < clt 10528 [,)cico 12594 lim supclsp 14665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-po 5369 df-so 5370 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-sup 8759 df-inf 8760 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-limsup 14666 |
This theorem is referenced by: (None) |
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