Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 42742 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
5 | 1, 2 | dmmeasal 42733 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
6 | meassle.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
7 | saldifcl2 42610 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆) | |
8 | 5, 6, 3, 7 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆) |
9 | 1, 2, 8 | meacl 42739 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) |
10 | 4, 9 | xadd0ge 41586 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
11 | meassle.ss | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
12 | undif 4429 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) | |
13 | 12 | biimpi 218 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
14 | 11, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
15 | 14 | fveq2d 6673 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = (𝑀‘𝐵)) |
16 | 15 | eqcomd 2827 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴)))) |
17 | disjdif 4420 | . . . . 5 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) |
19 | 1, 2, 3, 8, 18 | meadjun 42743 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
20 | 16, 19 | eqtr2d 2857 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴))) = (𝑀‘𝐵)) |
21 | 10, 20 | breqtrd 5091 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 class class class wbr 5065 dom cdm 5554 ‘cfv 6354 (class class class)co 7155 ≤ cle 10675 +𝑒 cxad 12504 SAlgcsalg 42592 Meascmea 42730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-disj 5031 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-xadd 12507 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 df-salg 42593 df-sumge0 42644 df-mea 42731 |
This theorem is referenced by: meaunle 42745 meaiunlelem 42749 meassre 42758 meaiuninclem 42761 meaiuninc3v 42765 meaiininclem 42767 vonioolem2 42962 |
Copyright terms: Public domain | W3C validator |