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Mirrors > Home > MPE Home > Th. List > limsuplt | 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 |
---|---|
limsuplt | ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑗 ∈ ℝ (𝐺‘𝑗) < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupval.1 | . . . . 5 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
2 | 1 | limsuple 15511 | . . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗))) |
3 | 2 | notbid 318 | . . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (¬ 𝐴 ≤ (lim sup‘𝐹) ↔ ¬ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗))) |
4 | rexnal 3098 | . . 3 ⊢ (∃𝑗 ∈ ℝ ¬ 𝐴 ≤ (𝐺‘𝑗) ↔ ¬ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗)) | |
5 | 3, 4 | bitr4di 289 | . 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (¬ 𝐴 ≤ (lim sup‘𝐹) ↔ ∃𝑗 ∈ ℝ ¬ 𝐴 ≤ (𝐺‘𝑗))) |
6 | simp2 1136 | . . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐹:𝐵⟶ℝ*) | |
7 | reex 11244 | . . . . . . 7 ⊢ ℝ ∈ V | |
8 | 7 | ssex 5327 | . . . . . 6 ⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V) |
9 | 8 | 3ad2ant1 1132 | . . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐵 ∈ V) |
10 | xrex 13027 | . . . . . 6 ⊢ ℝ* ∈ V | |
11 | 10 | a1i 11 | . . . . 5 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → ℝ* ∈ V) |
12 | fex2 7957 | . . . . 5 ⊢ ((𝐹:𝐵⟶ℝ* ∧ 𝐵 ∈ V ∧ ℝ* ∈ V) → 𝐹 ∈ V) | |
13 | 6, 9, 11, 12 | syl3anc 1370 | . . . 4 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐹 ∈ V) |
14 | limsupcl 15506 | . . . 4 ⊢ (𝐹 ∈ V → (lim sup‘𝐹) ∈ ℝ*) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (lim sup‘𝐹) ∈ ℝ*) |
16 | simp3 1137 | . . 3 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → 𝐴 ∈ ℝ*) | |
17 | xrltnle 11326 | . . 3 ⊢ (((lim sup‘𝐹) ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → ((lim sup‘𝐹) < 𝐴 ↔ ¬ 𝐴 ≤ (lim sup‘𝐹))) | |
18 | 15, 16, 17 | syl2anc 584 | . 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → ((lim sup‘𝐹) < 𝐴 ↔ ¬ 𝐴 ≤ (lim sup‘𝐹))) |
19 | 1 | limsupgf 15508 | . . . . 5 ⊢ 𝐺:ℝ⟶ℝ* |
20 | 19 | ffvelcdmi 7103 | . . . 4 ⊢ (𝑗 ∈ ℝ → (𝐺‘𝑗) ∈ ℝ*) |
21 | xrltnle 11326 | . . . 4 ⊢ (((𝐺‘𝑗) ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → ((𝐺‘𝑗) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐺‘𝑗))) | |
22 | 20, 16, 21 | syl2anr 597 | . . 3 ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) ∧ 𝑗 ∈ ℝ) → ((𝐺‘𝑗) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐺‘𝑗))) |
23 | 22 | rexbidva 3175 | . 2 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (∃𝑗 ∈ ℝ (𝐺‘𝑗) < 𝐴 ↔ ∃𝑗 ∈ ℝ ¬ 𝐴 ≤ (𝐺‘𝑗))) |
24 | 5, 18, 23 | 3bitr4d 311 | 1 ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑗 ∈ ℝ (𝐺‘𝑗) < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 “ cima 5692 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 supcsup 9478 ℝcr 11152 +∞cpnf 11290 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 [,)cico 13386 lim supclsp 15503 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-limsup 15504 |
This theorem is referenced by: limsupgre 15514 limsuplt2 45709 |
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