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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 30872 | . . . . . . 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 30808 | . . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆) | |
7 | 3, 4, 5, 6 | syl3anc 1439 | . . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆) |
8 | measvxrge0 30880 | . . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) | |
9 | 1, 7, 8 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) |
10 | elxrge0 12595 | . . . . 5 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ* ∧ 0 ≤ (𝑀‘(𝐵 ∖ 𝐴)))) | |
11 | 10 | simprbi 492 | . . . 4 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) → 0 ≤ (𝑀‘(𝐵 ∖ 𝐴))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝜑 → 0 ≤ (𝑀‘(𝐵 ∖ 𝐴))) |
13 | measvxrge0 30880 | . . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | |
14 | 1, 5, 13 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
15 | elxrge0 12595 | . . . . . 6 ⊢ ((𝑀‘𝐴) ∈ (0[,]+∞) ↔ ((𝑀‘𝐴) ∈ ℝ* ∧ 0 ≤ (𝑀‘𝐴))) | |
16 | 15 | simplbi 493 | . . . . 5 ⊢ ((𝑀‘𝐴) ∈ (0[,]+∞) → (𝑀‘𝐴) ∈ ℝ*) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
18 | 10 | simplbi 493 | . . . . 5 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ*) |
19 | 9, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ*) |
20 | xraddge02 30100 | . . . 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 579 | . . 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 4583 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆) | |
24 | 5, 7, 23 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆) |
25 | prex 5141 | . . . . . 6 ⊢ {𝐴, (𝐵 ∖ 𝐴)} ∈ V | |
26 | 25 | elpw 4384 | . . . . 5 ⊢ ({𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆 ↔ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆) |
27 | 24, 26 | sylibr 226 | . . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆) |
28 | prct 30072 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω) | |
29 | 5, 7, 28 | syl2anc 579 | . . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω) |
30 | disjdifprg 29965 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦) | |
31 | 5, 4, 30 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦) |
32 | prcom 4498 | . . . . . . 7 ⊢ {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)} | |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)}) |
34 | 33 | disjeq1d 4862 | . . . . 5 ⊢ (𝜑 → (Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦 ↔ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)) |
35 | 31, 34 | mpbid 224 | . . . 4 ⊢ (𝜑 → Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦) |
36 | measvun 30884 | . . . 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 1442 | . . 3 ⊢ (𝜑 → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦)) |
38 | uniprg 4685 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴))) | |
39 | 5, 7, 38 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴))) |
40 | measssd.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
41 | undif 4272 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) | |
42 | 40, 41 | sylib 210 | . . . . 5 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
43 | 39, 42 | eqtrd 2813 | . . . 4 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵) |
44 | 43 | fveq2d 6450 | . . 3 ⊢ (𝜑 → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = (𝑀‘𝐵)) |
45 | fveq2 6446 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑀‘𝑦) = (𝑀‘𝐴)) | |
46 | 45 | adantl 475 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑀‘𝑦) = (𝑀‘𝐴)) |
47 | fveq2 6446 | . . . . 5 ⊢ (𝑦 = (𝐵 ∖ 𝐴) → (𝑀‘𝑦) = (𝑀‘(𝐵 ∖ 𝐴))) | |
48 | 47 | adantl 475 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐵 ∖ 𝐴)) → (𝑀‘𝑦) = (𝑀‘(𝐵 ∖ 𝐴))) |
49 | eqimss 3875 | . . . . . . . . . 10 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 ⊆ (𝐵 ∖ 𝐴)) | |
50 | ssdifeq0 4274 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅) | |
51 | 49, 50 | sylib 210 | . . . . . . . . 9 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 = ∅) |
52 | 51 | adantl 475 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → 𝐴 = ∅) |
53 | 52 | fveq2d 6450 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘𝐴) = (𝑀‘∅)) |
54 | measvnul 30881 | . . . . . . . . 9 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0) | |
55 | 1, 54 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘∅) = 0) |
56 | 55 | adantr 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘∅) = 0) |
57 | 53, 56 | eqtrd 2813 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘𝐴) = 0) |
58 | 57 | orcd 862 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → ((𝑀‘𝐴) = 0 ∨ (𝑀‘𝐴) = +∞)) |
59 | 58 | ex 403 | . . . 4 ⊢ (𝜑 → (𝐴 = (𝐵 ∖ 𝐴) → ((𝑀‘𝐴) = 0 ∨ (𝑀‘𝐴) = +∞))) |
60 | 46, 48, 5, 7, 14, 9, 59 | esumpr2 30741 | . . 3 ⊢ (𝜑 → Σ*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
61 | 37, 44, 60 | 3eqtr3d 2821 | . 2 ⊢ (𝜑 → (𝑀‘𝐵) = ((𝑀‘𝐴) +𝑒 (𝑀‘(𝐵 ∖ 𝐴)))) |
62 | 22, 61 | breqtrrd 4914 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2106 ∖ cdif 3788 ∪ cun 3789 ⊆ wss 3791 ∅c0 4140 𝒫 cpw 4378 {cpr 4399 ∪ cuni 4671 Disj wdisj 4854 class class class wbr 4886 ran crn 5356 ‘cfv 6135 (class class class)co 6922 ωcom 7343 ≼ cdom 8239 0cc0 10272 +∞cpnf 10408 ℝ*cxr 10410 ≤ cle 10412 +𝑒 cxad 12255 [,]cicc 12490 Σ*cesum 30701 sigAlgebracsiga 30782 measurescmeas 30870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-ac2 9620 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-disj 4855 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-acn 9101 df-ac 9272 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-fac 13379 df-bc 13408 df-hash 13436 df-shft 14214 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 df-ef 15200 df-sin 15202 df-cos 15203 df-pi 15205 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-ordt 16547 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-ps 17586 df-tsr 17587 df-plusf 17627 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-subrg 19170 df-abv 19209 df-lmod 19257 df-scaf 19258 df-sra 19569 df-rgmod 19570 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-tmd 22284 df-tgp 22285 df-tsms 22338 df-trg 22371 df-xms 22533 df-ms 22534 df-tms 22535 df-nm 22795 df-ngp 22796 df-nrg 22798 df-nlm 22799 df-ii 23088 df-cncf 23089 df-limc 24067 df-dv 24068 df-log 24740 df-esum 30702 df-siga 30783 df-meas 30871 |
This theorem is referenced by: measiun 30893 aean 30919 sibfinima 31013 prob01 31088 probinc 31096 probmeasb 31105 cndprob01 31110 dstfrvinc 31151 |
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