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Mirrors > Home > MPE Home > Th. List > Mathboxes > measssd | Structured version Visualization version GIF version |
Description: A measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 28-Dec-2016.) |
Ref | Expression |
---|---|
measssd.1 | ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) |
measssd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
measssd.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
measssd.4 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
measssd | ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | measssd.1 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) | |
2 | measbase 31908 | . . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
4 | measssd.3 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
5 | measssd.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
6 | difelsiga 31844 | . . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆) | |
7 | 3, 4, 5, 6 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆) |
8 | measvxrge0 31916 | . . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) | |
9 | 1, 7, 8 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) |
10 | elxrge0 13074 | . . . . 5 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ* ∧ 0 ≤ (𝑀‘(𝐵 ∖ 𝐴)))) | |
11 | 10 | simprbi 500 | . . . 4 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) → 0 ≤ (𝑀‘(𝐵 ∖ 𝐴))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑀‘(𝐵 ∖ 𝐴))) |
13 | measvxrge0 31916 | . . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | |
14 | 1, 5, 13 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
15 | elxrge0 13074 | . . . . . 6 ⊢ ((𝑀‘𝐴) ∈ (0[,]+∞) ↔ ((𝑀‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝐴))) | |
16 | 15 | simplbi 501 | . . . . 5 ⊢ ((𝑀‘𝐴) ∈ (0[,]+∞) → (𝑀‘𝐴) ∈ ℝ*) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
18 | 10 | simplbi 501 | . . . . 5 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ*) |
19 | 9, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ*) |
20 | xraddge02 30830 | . . . 4 ⊢ (((𝑀‘𝐴) ∈ ℝ* ∧ (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ*) → (0 ≤ (𝑀‘(𝐵 ∖ 𝐴)) → (𝑀‘𝐴) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴))))) | |
21 | 17, 19, 20 | syl2anc 587 | . . 3 ⊢ (𝜑 → (0 ≤ (𝑀‘(𝐵 ∖ 𝐴)) → (𝑀‘𝐴) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴))))) |
22 | 12, 21 | mpd 15 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
23 | prssi 4750 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆) | |
24 | 5, 7, 23 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆) |
25 | prex 5341 | . . . . . 6 ⊢ {𝐴, (𝐵 ∖ 𝐴)} ∈ V | |
26 | 25 | elpw 4533 | . . . . 5 ⊢ ({𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆 ↔ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆) |
27 | 24, 26 | sylibr 237 | . . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆) |
28 | prct 30800 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω) | |
29 | 5, 7, 28 | syl2anc 587 | . . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω) |
30 | disjdifprg 30664 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦) | |
31 | 5, 4, 30 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦) |
32 | prcom 4664 | . . . . . . 7 ⊢ {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)} | |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)}) |
34 | 33 | disjeq1d 5042 | . . . . 5 ⊢ (𝜑 → (Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦 ↔ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)) |
35 | 31, 34 | mpbid 235 | . . . 4 ⊢ (𝜑 → Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦) |
36 | measvun 31920 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ {𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆 ∧ ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)) → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦)) | |
37 | 1, 27, 29, 35, 36 | syl112anc 1376 | . . 3 ⊢ (𝜑 → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦)) |
38 | uniprg 4852 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴))) | |
39 | 5, 7, 38 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴))) |
40 | measssd.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
41 | undif 4412 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) | |
42 | 40, 41 | sylib 221 | . . . . 5 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
43 | 39, 42 | eqtrd 2779 | . . . 4 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵) |
44 | 43 | fveq2d 6742 | . . 3 ⊢ (𝜑 → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = (𝑀‘𝐵)) |
45 | fveq2 6738 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑀‘𝑦) = (𝑀‘𝐴)) | |
46 | 45 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑀‘𝑦) = (𝑀‘𝐴)) |
47 | fveq2 6738 | . . . . 5 ⊢ (𝑦 = (𝐵 ∖ 𝐴) → (𝑀‘𝑦) = (𝑀‘(𝐵 ∖ 𝐴))) | |
48 | 47 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐵 ∖ 𝐴)) → (𝑀‘𝑦) = (𝑀‘(𝐵 ∖ 𝐴))) |
49 | eqimss 3973 | . . . . . . . . . 10 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 ⊆ (𝐵 ∖ 𝐴)) | |
50 | ssdifeq0 4414 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) | |
51 | 49, 50 | sylib 221 | . . . . . . . . 9 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 = ∅) |
52 | 51 | adantl 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → 𝐴 = ∅) |
53 | 52 | fveq2d 6742 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘𝐴) = (𝑀‘∅)) |
54 | measvnul 31917 | . . . . . . . . 9 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0) | |
55 | 1, 54 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘∅) = 0) |
56 | 55 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘∅) = 0) |
57 | 53, 56 | eqtrd 2779 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘𝐴) = 0) |
58 | 57 | orcd 873 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → ((𝑀‘𝐴) = 0 ∨ (𝑀‘𝐴) = +∞)) |
59 | 58 | ex 416 | . . . 4 ⊢ (𝜑 → (𝐴 = (𝐵 ∖ 𝐴) → ((𝑀‘𝐴) = 0 ∨ (𝑀‘𝐴) = +∞))) |
60 | 46, 48, 5, 7, 14, 9, 59 | esumpr2 31778 | . . 3 ⊢ (𝜑 → Σ*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
61 | 37, 44, 60 | 3eqtr3d 2787 | . 2 ⊢ (𝜑 → (𝑀‘𝐵) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
62 | 22, 61 | breqtrrd 5097 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 ∅c0 4253 𝒫 cpw 4529 {cpr 4559 ∪ cuni 4835 Disj wdisj 5034 class class class wbr 5069 ran crn 5569 ‘cfv 6400 (class class class)co 7234 ωcom 7665 ≼ cdom 8647 0cc0 10758 +∞cpnf 10893 ℝ*cxr 10895 ≤ cle 10897 +𝑒 cxad 12731 [,]cicc 12967 Σ*cesum 31738 sigAlgebracsiga 31819 measurescmeas 31906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-inf2 9285 ax-ac2 10106 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 ax-pre-sup 10836 ax-addf 10837 ax-mulf 10838 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-int 4876 df-iun 4922 df-iin 4923 df-disj 5035 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-se 5527 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-isom 6409 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-of 7490 df-om 7666 df-1st 7782 df-2nd 7783 df-supp 7927 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-1o 8225 df-2o 8226 df-er 8414 df-map 8533 df-pm 8534 df-ixp 8602 df-en 8650 df-dom 8651 df-sdom 8652 df-fin 8653 df-fsupp 9015 df-fi 9056 df-sup 9087 df-inf 9088 df-oi 9155 df-dju 9546 df-card 9584 df-acn 9587 df-ac 9759 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-div 11519 df-nn 11860 df-2 11922 df-3 11923 df-4 11924 df-5 11925 df-6 11926 df-7 11927 df-8 11928 df-9 11929 df-n0 12120 df-z 12206 df-dec 12323 df-uz 12468 df-q 12574 df-rp 12616 df-xneg 12733 df-xadd 12734 df-xmul 12735 df-ioo 12968 df-ioc 12969 df-ico 12970 df-icc 12971 df-fz 13125 df-fzo 13268 df-fl 13396 df-mod 13474 df-seq 13606 df-exp 13667 df-fac 13872 df-bc 13901 df-hash 13929 df-shft 14662 df-cj 14694 df-re 14695 df-im 14696 df-sqrt 14830 df-abs 14831 df-limsup 15064 df-clim 15081 df-rlim 15082 df-sum 15282 df-ef 15661 df-sin 15663 df-cos 15664 df-pi 15666 df-struct 16732 df-sets 16749 df-slot 16767 df-ndx 16777 df-base 16793 df-ress 16817 df-plusg 16847 df-mulr 16848 df-starv 16849 df-sca 16850 df-vsca 16851 df-ip 16852 df-tset 16853 df-ple 16854 df-ds 16856 df-unif 16857 df-hom 16858 df-cco 16859 df-rest 16959 df-topn 16960 df-0g 16978 df-gsum 16979 df-topgen 16980 df-pt 16981 df-prds 16984 df-ordt 17038 df-xrs 17039 df-qtop 17044 df-imas 17045 df-xps 17047 df-mre 17121 df-mrc 17122 df-acs 17124 df-ps 18104 df-tsr 18105 df-plusf 18145 df-mgm 18146 df-sgrp 18195 df-mnd 18206 df-mhm 18250 df-submnd 18251 df-grp 18400 df-minusg 18401 df-sbg 18402 df-mulg 18521 df-subg 18572 df-cntz 18743 df-cmn 19204 df-abl 19205 df-mgp 19537 df-ur 19549 df-ring 19596 df-cring 19597 df-subrg 19830 df-abv 19885 df-lmod 19933 df-scaf 19934 df-sra 20241 df-rgmod 20242 df-psmet 20387 df-xmet 20388 df-met 20389 df-bl 20390 df-mopn 20391 df-fbas 20392 df-fg 20393 df-cnfld 20396 df-top 21822 df-topon 21839 df-topsp 21861 df-bases 21874 df-cld 21947 df-ntr 21948 df-cls 21949 df-nei 22026 df-lp 22064 df-perf 22065 df-cn 22155 df-cnp 22156 df-haus 22243 df-tx 22490 df-hmeo 22683 df-fil 22774 df-fm 22866 df-flim 22867 df-flf 22868 df-tmd 23000 df-tgp 23001 df-tsms 23055 df-trg 23088 df-xms 23249 df-ms 23250 df-tms 23251 df-nm 23511 df-ngp 23512 df-nrg 23514 df-nlm 23515 df-ii 23805 df-cncf 23806 df-limc 24794 df-dv 24795 df-log 25476 df-esum 31739 df-siga 31820 df-meas 31907 |
This theorem is referenced by: measiun 31929 aean 31955 sibfinima 32049 prob01 32123 probinc 32131 probmeasb 32140 cndprob01 32145 dstfrvinc 32186 |
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