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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meassle | Structured version Visualization version GIF version |
Description: The measure of a set is greater than or equal to the measure of a subset, Property 112C (b) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
meassle.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meassle.x | ⊢ 𝑆 = dom 𝑀 |
meassle.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
meassle.b | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
meassle.ss | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
meassle | ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meassle.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
2 | meassle.x | . . . 4 ⊢ 𝑆 = dom 𝑀 | |
3 | meassle.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | 1, 2, 3 | meaxrcl 43100 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
5 | 1, 2 | dmmeasal 43091 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
6 | meassle.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
7 | saldifcl2 42968 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆) | |
8 | 5, 6, 3, 7 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆) |
9 | 1, 2, 8 | meacl 43097 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) |
10 | 4, 9 | xadd0ge 41952 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
11 | meassle.ss | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
12 | undif 4388 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) | |
13 | 12 | biimpi 219 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
14 | 11, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
15 | 14 | fveq2d 6649 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = (𝑀‘𝐵)) |
16 | 15 | eqcomd 2804 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴)))) |
17 | disjdif 4379 | . . . . 5 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) |
19 | 1, 2, 3, 8, 18 | meadjun 43101 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
20 | 16, 19 | eqtr2d 2834 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴))) = (𝑀‘𝐵)) |
21 | 10, 20 | breqtrd 5056 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ≤ cle 10665 +𝑒 cxad 12493 SAlgcsalg 42950 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: meaunle 43103 meaiunlelem 43107 meassre 43116 meaiuninclem 43119 meaiuninc3v 43123 meaiininclem 43125 vonioolem2 43320 |
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