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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > volmea | Structured version Visualization version GIF version |
Description: The Lebesgue 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 45734 | . . 3 ⊢ dom vol ∈ SAlg | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → dom vol ∈ SAlg) |
3 | volf 25471 | . . 3 ⊢ vol:dom vol⟶(0[,]+∞) | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞)) |
5 | vol0 45347 | . . 3 ⊢ (vol‘∅) = 0 | |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → (vol‘∅) = 0) |
7 | simp1 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝜑) | |
8 | simp2 1135 | . . 3 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → 𝑒:ℕ⟶dom vol) | |
9 | fveq2 6897 | . . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑒‘𝑚) = (𝑒‘𝑛)) | |
10 | 9 | cbvdisjv 5124 | . . . . 5 ⊢ (Disj 𝑚 ∈ ℕ (𝑒‘𝑚) ↔ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) |
11 | 10 | biimpri 227 | . . . 4 ⊢ (Disj 𝑛 ∈ ℕ (𝑒‘𝑛) → Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) |
12 | 11 | 3ad2ant3 1133 | . . 3 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) |
13 | simp2 1135 | . . . 4 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) → 𝑒:ℕ⟶dom vol) | |
14 | 10 | biimpi 215 | . . . . 5 ⊢ (Disj 𝑚 ∈ ℕ (𝑒‘𝑚) → Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) |
15 | 14 | 3ad2ant3 1133 | . . . 4 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶dom vol ∧ Disj 𝑚 ∈ ℕ (𝑒‘𝑚)) → Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) |
16 | 13, 15 | voliunsge0 45861 | . . 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 1369 | . 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 45855 | 1 ⊢ (𝜑 → vol ∈ Meas) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∅c0 4323 ∪ ciun 4996 Disj wdisj 5113 ↦ cmpt 5231 dom cdm 5678 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 0cc0 11139 +∞cpnf 11276 ℕcn 12243 [,]cicc 13360 volcvol 25405 SAlgcsalg 45696 Σ^csumge0 45750 Meascmea 45837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cc 10459 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-oi 9534 df-dju 9925 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-q 12964 df-rp 13008 df-xadd 13126 df-ioo 13361 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-rlim 15466 df-sum 15666 df-xmet 21272 df-met 21273 df-ovol 25406 df-vol 25407 df-salg 45697 df-sumge0 45751 df-mea 45838 |
This theorem is referenced by: (None) |
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