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Mirrors > Home > MPE Home > Th. List > Mathboxes > measiun | Structured version Visualization version GIF version |
Description: A measure is sub-additive. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.) |
Ref | Expression |
---|---|
measiun.1 | ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) |
measiun.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
measiun.3 | ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ 𝑆) |
measiun.4 | ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐵) |
Ref | Expression |
---|---|
measiun | ⊢ (𝜑 → (𝑀‘𝐴) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 12983 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | measiun.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) | |
3 | measiun.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | measvxrge0 31839 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | |
5 | 2, 3, 4 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
6 | 1, 5 | sseldi 3885 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
7 | measbase 31831 | . . . . . 6 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
8 | 2, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
9 | measiun.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ 𝑆) | |
10 | 9 | ralrimiva 3095 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐵 ∈ 𝑆) |
11 | sigaclcu2 31754 | . . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑛 ∈ ℕ 𝐵 ∈ 𝑆) → ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) | |
12 | 8, 10, 11 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) |
13 | measvxrge0 31839 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ (0[,]+∞)) | |
14 | 2, 12, 13 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ (0[,]+∞)) |
15 | 1, 14 | sseldi 3885 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ ℝ*) |
16 | nnex 11801 | . . . 4 ⊢ ℕ ∈ V | |
17 | 2 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆)) |
18 | measvxrge0 31839 | . . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ (0[,]+∞)) | |
19 | 17, 9, 18 | syl2anc 587 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝐵) ∈ (0[,]+∞)) |
20 | 19 | ralrimiva 3095 | . . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑀‘𝐵) ∈ (0[,]+∞)) |
21 | nfcv 2897 | . . . . 5 ⊢ Ⅎ𝑛ℕ | |
22 | 21 | esumcl 31664 | . . . 4 ⊢ ((ℕ ∈ V ∧ ∀𝑛 ∈ ℕ (𝑀‘𝐵) ∈ (0[,]+∞)) → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ (0[,]+∞)) |
23 | 16, 20, 22 | sylancr 590 | . . 3 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ (0[,]+∞)) |
24 | 1, 23 | sseldi 3885 | . 2 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ ℝ*) |
25 | measiun.4 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐵) | |
26 | 2, 3, 12, 25 | measssd 31849 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘∪ 𝑛 ∈ ℕ 𝐵)) |
27 | nfcsb1v 3823 | . . . 4 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 | |
28 | csbeq1a 3812 | . . . 4 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵) | |
29 | eqidd 2737 | . . . . 5 ⊢ (𝜑 → ℕ = ℕ) | |
30 | 29 | orcd 873 | . . . 4 ⊢ (𝜑 → (ℕ = ℕ ∨ ℕ = (1..^𝑚))) |
31 | 27, 28, 30, 2, 9 | measiuns 31851 | . . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) = Σ*𝑛 ∈ ℕ(𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵))) |
32 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ ∈ V) |
33 | 8 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ∪ ran sigAlgebra) |
34 | nfv 1922 | . . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑 | |
35 | nfcv 2897 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘 | |
36 | 35 | nfel1 2913 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑘 ∈ ℕ |
37 | 27 | nfel1 2913 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆 |
38 | 36, 37 | nfim 1904 | . . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
39 | 34, 38 | nfim 1904 | . . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
40 | eleq1w 2813 | . . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ)) | |
41 | 28 | eleq1d 2815 | . . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐵 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
42 | 40, 41 | imbi12d 348 | . . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑛 ∈ ℕ → 𝐵 ∈ 𝑆) ↔ (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆))) |
43 | 42 | imbi2d 344 | . . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝜑 → (𝑛 ∈ ℕ → 𝐵 ∈ 𝑆)) ↔ (𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)))) |
44 | 9 | ex 416 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ → 𝐵 ∈ 𝑆)) |
45 | 39, 43, 44 | chvarfv 2240 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
46 | 45 | ralrimiv 3094 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
47 | fzossnn 13256 | . . . . . . . . . 10 ⊢ (1..^𝑛) ⊆ ℕ | |
48 | ssralv 3953 | . . . . . . . . . 10 ⊢ ((1..^𝑛) ⊆ ℕ → (∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆 → ∀𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) | |
49 | 47, 48 | ax-mp 5 | . . . . . . . . 9 ⊢ (∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆 → ∀𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
50 | sigaclfu2 31755 | . . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) | |
51 | 49, 50 | sylan2 596 | . . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
52 | 8, 46, 51 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
53 | 52 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
54 | difelsiga 31767 | . . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝑆 ∧ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ∈ 𝑆) | |
55 | 33, 9, 53, 54 | syl3anc 1373 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ∈ 𝑆) |
56 | measvxrge0 31839 | . . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ∈ 𝑆) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ∈ (0[,]+∞)) | |
57 | 17, 55, 56 | syl2anc 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ∈ (0[,]+∞)) |
58 | difssd 4033 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ⊆ 𝐵) | |
59 | 17, 55, 9, 58 | measssd 31849 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ (𝑀‘𝐵)) |
60 | 32, 57, 19, 59 | esumle 31692 | . . 3 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
61 | 31, 60 | eqbrtrd 5061 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
62 | 6, 15, 24, 26, 61 | xrletrd 12717 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 ⦋csb 3798 ∖ cdif 3850 ⊆ wss 3853 ∪ cuni 4805 ∪ ciun 4890 class class class wbr 5039 ran crn 5537 ‘cfv 6358 (class class class)co 7191 0cc0 10694 1c1 10695 +∞cpnf 10829 ℝ*cxr 10831 ≤ cle 10833 ℕcn 11795 [,]cicc 12903 ..^cfzo 13203 Σ*cesum 31661 sigAlgebracsiga 31742 measurescmeas 31829 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-ac2 10042 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-disj 5005 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-dju 9482 df-card 9520 df-acn 9523 df-ac 9695 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-fac 13805 df-bc 13834 df-hash 13862 df-shft 14595 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-limsup 14997 df-clim 15014 df-rlim 15015 df-sum 15215 df-ef 15592 df-sin 15594 df-cos 15595 df-pi 15597 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-ordt 16960 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-ps 18026 df-tsr 18027 df-plusf 18067 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-subrg 19752 df-abv 19807 df-lmod 19855 df-scaf 19856 df-sra 20163 df-rgmod 20164 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-fbas 20314 df-fg 20315 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-lp 21987 df-perf 21988 df-cn 22078 df-cnp 22079 df-haus 22166 df-tx 22413 df-hmeo 22606 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-tmd 22923 df-tgp 22924 df-tsms 22978 df-trg 23011 df-xms 23172 df-ms 23173 df-tms 23174 df-nm 23434 df-ngp 23435 df-nrg 23437 df-nlm 23438 df-ii 23728 df-cncf 23729 df-limc 24717 df-dv 24718 df-log 25399 df-esum 31662 df-siga 31743 df-meas 31830 |
This theorem is referenced by: (None) |
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