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 43575 | . . 3 ⊢ dom vol ∈ SAlg | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → dom vol ∈ SAlg) |
3 | volf 24439 | . . 3 ⊢ vol:dom vol⟶(0[,]+∞) | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞)) |
5 | vol0 43190 | . . 3 ⊢ (vol‘∅) = 0 | |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → (vol‘∅) = 0) |
7 | simp1 1138 | . . 3 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝜑) | |
8 | simp2 1139 | . . 3 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝑒:ℕ⟶dom vol) | |
9 | fveq2 6726 | . . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑒‘𝑚) = (𝑒‘𝑛)) | |
10 | 9 | cbvdisjv 5038 | . . . . 5 ⊢ (Disj 𝑚 ∈ ℕ (𝑒‘𝑚) ↔ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) |
11 | 10 | biimpri 231 | . . . 4 ⊢ (Disj 𝑛 ∈ ℕ (𝑒‘𝑛) → Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) |
12 | 11 | 3ad2ant3 1137 | . . 3 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) |
13 | simp2 1139 | . . . 4 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) → 𝑒:ℕ⟶dom vol) | |
14 | 10 | biimpi 219 | . . . . 5 ⊢ (Disj 𝑚 ∈ ℕ (𝑒‘𝑚) → Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) |
15 | 14 | 3ad2ant3 1137 | . . . 4 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) → Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) |
16 | 13, 15 | voliunsge0 43701 | . . 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 1373 | . 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 43695 | 1 ⊢ (𝜑 → vol ∈ Meas) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ∅c0 4246 ∪ ciun 4913 Disj wdisj 5027 ↦ cmpt 5144 dom cdm 5560 ⟶wf 6385 ‘cfv 6389 (class class class)co 7222 0cc0 10742 +∞cpnf 10877 ℕcn 11843 [,]cicc 12951 volcvol 24373 SAlgcsalg 43539 Σ^csumge0 43590 Meascmea 43677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-inf2 9269 ax-cc 10062 ax-cnex 10798 ax-resscn 10799 ax-1cn 10800 ax-icn 10801 ax-addcl 10802 ax-addrcl 10803 ax-mulcl 10804 ax-mulrcl 10805 ax-mulcom 10806 ax-addass 10807 ax-mulass 10808 ax-distr 10809 ax-i2m1 10810 ax-1ne0 10811 ax-1rid 10812 ax-rnegex 10813 ax-rrecex 10814 ax-cnre 10815 ax-pre-lttri 10816 ax-pre-lttrn 10817 ax-pre-ltadd 10818 ax-pre-mulgt0 10819 ax-pre-sup 10820 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-int 4869 df-iun 4915 df-disj 5028 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-se 5519 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-isom 6398 df-riota 7179 df-ov 7225 df-oprab 7226 df-mpo 7227 df-of 7478 df-om 7654 df-1st 7770 df-2nd 7771 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-1o 8211 df-2o 8212 df-er 8400 df-map 8519 df-pm 8520 df-en 8636 df-dom 8637 df-sdom 8638 df-fin 8639 df-sup 9071 df-inf 9072 df-oi 9139 df-dju 9530 df-card 9568 df-pnf 10882 df-mnf 10883 df-xr 10884 df-ltxr 10885 df-le 10886 df-sub 11077 df-neg 11078 df-div 11503 df-nn 11844 df-2 11906 df-3 11907 df-n0 12104 df-z 12190 df-uz 12452 df-q 12558 df-rp 12600 df-xadd 12718 df-ioo 12952 df-ico 12954 df-icc 12955 df-fz 13109 df-fzo 13252 df-fl 13380 df-seq 13588 df-exp 13649 df-hash 13910 df-cj 14675 df-re 14676 df-im 14677 df-sqrt 14811 df-abs 14812 df-clim 15062 df-rlim 15063 df-sum 15263 df-xmet 20369 df-met 20370 df-ovol 24374 df-vol 24375 df-salg 43540 df-sumge0 43591 df-mea 43678 |
This theorem is referenced by: (None) |
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