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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 32796 | . . . . . . 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 32732 | . . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆) | |
7 | 3, 4, 5, 6 | syl3anc 1371 | . . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆) |
8 | measvxrge0 32804 | . . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) | |
9 | 1, 7, 8 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) |
10 | elxrge0 13374 | . . . . 5 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ* ∧ 0 ≤ (𝑀‘(𝐵 ∖ 𝐴)))) | |
11 | 10 | simprbi 497 | . . . 4 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) → 0 ≤ (𝑀‘(𝐵 ∖ 𝐴))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑀‘(𝐵 ∖ 𝐴))) |
13 | measvxrge0 32804 | . . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | |
14 | 1, 5, 13 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
15 | elxrge0 13374 | . . . . . 6 ⊢ ((𝑀‘𝐴) ∈ (0[,]+∞) ↔ ((𝑀‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝐴))) | |
16 | 15 | simplbi 498 | . . . . 5 ⊢ ((𝑀‘𝐴) ∈ (0[,]+∞) → (𝑀‘𝐴) ∈ ℝ*) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
18 | 10 | simplbi 498 | . . . . 5 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ*) |
19 | 9, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ*) |
20 | xraddge02 31661 | . . . 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 584 | . . 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 4781 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆) | |
24 | 5, 7, 23 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆) |
25 | prex 5389 | . . . . . 6 ⊢ {𝐴, (𝐵 ∖ 𝐴)} ∈ V | |
26 | 25 | elpw 4564 | . . . . 5 ⊢ ({𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆 ↔ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆) |
27 | 24, 26 | sylibr 233 | . . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆) |
28 | prct 31631 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω) | |
29 | 5, 7, 28 | syl2anc 584 | . . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω) |
30 | disjdifprg 31493 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦) | |
31 | 5, 4, 30 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦) |
32 | prcom 4693 | . . . . . . 7 ⊢ {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)} | |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)}) |
34 | 33 | disjeq1d 5078 | . . . . 5 ⊢ (𝜑 → (Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦 ↔ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)) |
35 | 31, 34 | mpbid 231 | . . . 4 ⊢ (𝜑 → Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦) |
36 | measvun 32808 | . . . 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 1374 | . . 3 ⊢ (𝜑 → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦)) |
38 | uniprg 4882 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴))) | |
39 | 5, 7, 38 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴))) |
40 | measssd.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
41 | undif 4441 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) | |
42 | 40, 41 | sylib 217 | . . . . 5 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
43 | 39, 42 | eqtrd 2776 | . . . 4 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵) |
44 | 43 | fveq2d 6846 | . . 3 ⊢ (𝜑 → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = (𝑀‘𝐵)) |
45 | fveq2 6842 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑀‘𝑦) = (𝑀‘𝐴)) | |
46 | 45 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑀‘𝑦) = (𝑀‘𝐴)) |
47 | fveq2 6842 | . . . . 5 ⊢ (𝑦 = (𝐵 ∖ 𝐴) → (𝑀‘𝑦) = (𝑀‘(𝐵 ∖ 𝐴))) | |
48 | 47 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐵 ∖ 𝐴)) → (𝑀‘𝑦) = (𝑀‘(𝐵 ∖ 𝐴))) |
49 | eqimss 4000 | . . . . . . . . . 10 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 ⊆ (𝐵 ∖ 𝐴)) | |
50 | ssdifeq0 4444 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) | |
51 | 49, 50 | sylib 217 | . . . . . . . . 9 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 = ∅) |
52 | 51 | adantl 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → 𝐴 = ∅) |
53 | 52 | fveq2d 6846 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘𝐴) = (𝑀‘∅)) |
54 | measvnul 32805 | . . . . . . . . 9 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0) | |
55 | 1, 54 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘∅) = 0) |
56 | 55 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘∅) = 0) |
57 | 53, 56 | eqtrd 2776 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘𝐴) = 0) |
58 | 57 | orcd 871 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → ((𝑀‘𝐴) = 0 ∨ (𝑀‘𝐴) = +∞)) |
59 | 58 | ex 413 | . . . 4 ⊢ (𝜑 → (𝐴 = (𝐵 ∖ 𝐴) → ((𝑀‘𝐴) = 0 ∨ (𝑀‘𝐴) = +∞))) |
60 | 46, 48, 5, 7, 14, 9, 59 | esumpr2 32666 | . . 3 ⊢ (𝜑 → Σ*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
61 | 37, 44, 60 | 3eqtr3d 2784 | . 2 ⊢ (𝜑 → (𝑀‘𝐵) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
62 | 22, 61 | breqtrrd 5133 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∖ cdif 3907 ∪ cun 3908 ⊆ wss 3910 ∅c0 4282 𝒫 cpw 4560 {cpr 4588 ∪ cuni 4865 Disj wdisj 5070 class class class wbr 5105 ran crn 5634 ‘cfv 6496 (class class class)co 7357 ωcom 7802 ≼ cdom 8881 0cc0 11051 +∞cpnf 11186 ℝ*cxr 11188 ≤ cle 11190 +𝑒 cxad 13031 [,]cicc 13267 Σ*cesum 32626 sigAlgebracsiga 32707 measurescmeas 32794 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-ac2 10399 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-ac 10052 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-ordt 17383 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-ps 18455 df-tsr 18456 df-plusf 18496 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-subrg 20220 df-abv 20276 df-lmod 20324 df-scaf 20325 df-sra 20633 df-rgmod 20634 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-tmd 23423 df-tgp 23424 df-tsms 23478 df-trg 23511 df-xms 23673 df-ms 23674 df-tms 23675 df-nm 23938 df-ngp 23939 df-nrg 23941 df-nlm 23942 df-ii 24240 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 df-esum 32627 df-siga 32708 df-meas 32795 |
This theorem is referenced by: measiun 32817 aean 32843 sibfinima 32939 prob01 33013 probinc 33021 probmeasb 33030 cndprob01 33035 dstfrvinc 33076 |
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