Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > volmea | Structured version Visualization version GIF version |
Description: The Lebeasgue measure on the Reals is actually a measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
volmea | ⊢ (𝜑 → vol ∈ Meas) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmvolsal 40564 | . . 3 ⊢ dom vol ∈ SAlg | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → dom vol ∈ SAlg) |
3 | volf 23297 | . . 3 ⊢ vol:dom vol⟶(0[,]+∞) | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞)) |
5 | vol0 40175 | . . 3 ⊢ (vol‘∅) = 0 | |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → (vol‘∅) = 0) |
7 | simp1 1061 | . . 3 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝜑) | |
8 | simp2 1062 | . . 3 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝑒:ℕ⟶dom vol) | |
9 | fveq2 6191 | . . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑒‘𝑚) = (𝑒‘𝑛)) | |
10 | 9 | cbvdisjv 4631 | . . . . 5 ⊢ (Disj 𝑚 ∈ ℕ (𝑒‘𝑚) ↔ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) |
11 | 10 | biimpri 218 | . . . 4 ⊢ (Disj 𝑛 ∈ ℕ (𝑒‘𝑛) → Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) |
12 | 11 | 3ad2ant3 1084 | . . 3 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) |
13 | simp2 1062 | . . . 4 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) → 𝑒:ℕ⟶dom vol) | |
14 | 10 | biimpi 206 | . . . . 5 ⊢ (Disj 𝑚 ∈ ℕ (𝑒‘𝑚) → Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) |
15 | 14 | 3ad2ant3 1084 | . . . 4 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) → Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) |
16 | 13, 15 | voliunsge0 40690 | . . 3 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) → (vol‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝑒‘𝑛))))) |
17 | 7, 8, 12, 16 | syl3anc 1326 | . 2 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (vol‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝑒‘𝑛))))) |
18 | 2, 4, 6, 17 | ismeannd 40684 | 1 ⊢ (𝜑 → vol ∈ Meas) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∅c0 3915 ∪ ciun 4520 Disj wdisj 4620 ↦ cmpt 4729 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 +∞cpnf 10071 ℕcn 11020 [,]cicc 12178 volcvol 23232 SAlgcsalg 40528 Σ^csumge0 40579 Meascmea 40666 |
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-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-int 4476 df-iun 4522 df-disj 4621 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-xmet 19739 df-met 19740 df-ovol 23233 df-vol 23234 df-salg 40529 df-sumge0 40580 df-mea 40667 |
This theorem is referenced by: (None) |
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