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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meassre | Structured version Visualization version GIF version |
Description: If the measure of a measurable set is real, then the measure of any of its measurable subsets is real. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
meassre.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meassre.a | ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) |
meassre.r | ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ) |
meassre.s | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
meassre.b | ⊢ (𝜑 → 𝐵 ∈ dom 𝑀) |
Ref | Expression |
---|---|
meassre | ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rge0ssre 12834 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 0xr 10677 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
4 | pnfxr 10684 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
6 | meassre.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
7 | eqid 2798 | . . . 4 ⊢ dom 𝑀 = dom 𝑀 | |
8 | meassre.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑀) | |
9 | 6, 7, 8 | meaxrcl 43100 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ*) |
10 | 6, 8 | meage0 43114 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑀‘𝐵)) |
11 | meassre.r | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ) | |
12 | 11 | rexrd 10680 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
13 | meassre.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) | |
14 | meassre.s | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
15 | 6, 7, 8, 13, 14 | meassle 43102 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ≤ (𝑀‘𝐴)) |
16 | 11 | ltpnfd 12504 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) < +∞) |
17 | 9, 12, 5, 15, 16 | xrlelttrd 12541 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) < +∞) |
18 | 3, 5, 9, 10, 17 | elicod 12775 | . 2 ⊢ (𝜑 → (𝑀‘𝐵) ∈ (0[,)+∞)) |
19 | 1, 18 | sseldi 3913 | 1 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ⊆ wss 3881 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 +∞cpnf 10661 ℝ*cxr 10663 [,)cico 12728 Meascmea 43088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-xadd 12496 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-salg 42951 df-sumge0 43002 df-mea 43089 |
This theorem is referenced by: meadif 43118 meaiininclem 43125 vonioolem2 43320 |
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