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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ovn0val | Structured version Visualization version GIF version | ||
| Description: The Lebesgue outer measure (for the zero dimensional space of reals) of every subset is zero. (Contributed by Glauco Siliprandi, 11-Oct-2020.) | 
| Ref | Expression | 
|---|---|
| ovn0val.1 | ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m ∅)) | 
| Ref | Expression | 
|---|---|
| ovn0val | ⊢ (𝜑 → ((voln*‘∅)‘𝐴) = 0) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0fi 9082 | . . . 4 ⊢ ∅ ∈ Fin | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ Fin) | 
| 3 | ovn0val.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m ∅)) | |
| 4 | eqid 2737 | . . 3 ⊢ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m ∅) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ ∅ (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m ∅) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ ∅ (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} | |
| 5 | 2, 3, 4 | ovnval2 46560 | . 2 ⊢ (𝜑 → ((voln*‘∅)‘𝐴) = if(∅ = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m ∅) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ ∅ (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < ))) | 
| 6 | eqid 2737 | . . . 4 ⊢ ∅ = ∅ | |
| 7 | iftrue 4531 | . . . 4 ⊢ (∅ = ∅ → if(∅ = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m ∅) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ ∅ (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < )) = 0) | |
| 8 | 6, 7 | ax-mp 5 | . . 3 ⊢ if(∅ = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m ∅) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ ∅ (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < )) = 0 | 
| 9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → if(∅ = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m ∅) ↑m ℕ)(𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ ∅ (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < )) = 0) | 
| 10 | 5, 9 | eqtrd 2777 | 1 ⊢ (𝜑 → ((voln*‘∅)‘𝐴) = 0) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 ⊆ wss 3951 ∅c0 4333 ifcif 4525 ∪ ciun 4991 ↦ cmpt 5225 × cxp 5683 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Xcixp 8937 Fincfn 8985 infcinf 9481 ℝcr 11154 0cc0 11155 ℝ*cxr 11294 < clt 11295 ℕcn 12266 [,)cico 13389 ∏cprod 15939 volcvol 25498 Σ^csumge0 46377 voln*covoln 46551 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-seq 14043 df-prod 15940 df-ovoln 46552 | 
| This theorem is referenced by: ovnssle 46576 ovn02 46583 ovnsubadd 46587 ovnhoi 46618 ovnlecvr2 46625 von0val 46686 | 
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