Mathbox for Glauco Siliprandi |
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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 | 0fin 8188 | . . . 4 ⊢ ∅ ∈ Fin | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ Fin) |
3 | ovn0val.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 ∅)) | |
4 | eqid 2622 | . . 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 40759 | . 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 2622 | . . . 4 ⊢ ∅ = ∅ | |
7 | iftrue 4092 | . . . 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 2656 | 1 ⊢ (𝜑 → ((voln*‘∅)‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 {crab 2916 ⊆ wss 3574 ∅c0 3915 ifcif 4086 ∪ ciun 4520 ↦ cmpt 4729 × cxp 5112 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Xcixp 7908 Fincfn 7955 infcinf 8347 ℝcr 9935 0cc0 9936 ℝ*cxr 10073 < clt 10074 ℕcn 11020 [,)cico 12177 ∏cprod 14635 volcvol 23232 Σ^csumge0 40579 voln*covoln 40750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-i2m1 10004 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-seq 12802 df-prod 14636 df-ovoln 40751 |
This theorem is referenced by: ovnssle 40775 ovn02 40782 ovnsubadd 40786 ovnhoi 40817 ovnlecvr2 40824 von0val 40885 |
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