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 12838 | . 2 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | 0xr 10681 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
4 | pnfxr 10688 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) |
6 | meassre.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
7 | eqid 2820 | . . . 4 ⊢ dom 𝑀 = dom 𝑀 | |
8 | meassre.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑀) | |
9 | 6, 7, 8 | meaxrcl 42824 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ*) |
10 | 6, 8 | meage0 42838 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑀‘𝐵)) |
11 | meassre.r | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ) | |
12 | 11 | rexrd 10684 | . . . 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 42826 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ≤ (𝑀‘𝐴)) |
16 | 11 | ltpnfd 12510 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) < +∞) |
17 | 9, 12, 5, 15, 16 | xrlelttrd 12547 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) < +∞) |
18 | 3, 5, 9, 10, 17 | elicod 12781 | . 2 ⊢ (𝜑 → (𝑀‘𝐵) ∈ (0[,)+∞)) |
19 | 1, 18 | sseldi 3958 | 1 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ⊆ wss 3929 dom cdm 5548 ‘cfv 6348 (class class class)co 7149 ℝcr 10529 0cc0 10530 +∞cpnf 10665 ℝ*cxr 10667 [,)cico 12734 Meascmea 42812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-disj 5025 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-xadd 12502 df-ico 12738 df-icc 12739 df-fz 12890 df-fzo 13031 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14451 df-re 14452 df-im 14453 df-sqrt 14587 df-abs 14588 df-clim 14838 df-sum 15036 df-salg 42675 df-sumge0 42726 df-mea 42813 |
This theorem is referenced by: meadif 42842 meaiininclem 42849 vonioolem2 43044 |
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