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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 12822 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | measiun.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) | |
3 | measiun.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | measvxrge0 31468 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | |
5 | 2, 3, 4 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
6 | 1, 5 | sseldi 3968 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
7 | measbase 31460 | . . . . . 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 3185 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐵 ∈ 𝑆) |
11 | sigaclcu2 31383 | . . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑛 ∈ ℕ 𝐵 ∈ 𝑆) → ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) | |
12 | 8, 10, 11 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) |
13 | measvxrge0 31468 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ (0[,]+∞)) | |
14 | 2, 12, 13 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ (0[,]+∞)) |
15 | 1, 14 | sseldi 3968 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ ℝ*) |
16 | nnex 11647 | . . . 4 ⊢ ℕ ∈ V | |
17 | 2 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆)) |
18 | measvxrge0 31468 | . . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ (0[,]+∞)) | |
19 | 17, 9, 18 | syl2anc 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝐵) ∈ (0[,]+∞)) |
20 | 19 | ralrimiva 3185 | . . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑀‘𝐵) ∈ (0[,]+∞)) |
21 | nfcv 2980 | . . . . 5 ⊢ Ⅎ𝑛ℕ | |
22 | 21 | esumcl 31293 | . . . 4 ⊢ ((ℕ ∈ V ∧ ∀𝑛 ∈ ℕ (𝑀‘𝐵) ∈ (0[,]+∞)) → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ (0[,]+∞)) |
23 | 16, 20, 22 | sylancr 589 | . . 3 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ (0[,]+∞)) |
24 | 1, 23 | sseldi 3968 | . 2 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ ℝ*) |
25 | measiun.4 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐵) | |
26 | 2, 3, 12, 25 | measssd 31478 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘∪ 𝑛 ∈ ℕ 𝐵)) |
27 | nfcsb1v 3910 | . . . 4 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 | |
28 | csbeq1a 3900 | . . . 4 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵) | |
29 | eqidd 2825 | . . . . 5 ⊢ (𝜑 → ℕ = ℕ) | |
30 | 29 | orcd 869 | . . . 4 ⊢ (𝜑 → (ℕ = ℕ ∨ ℕ = (1..^𝑚))) |
31 | 27, 28, 30, 2, 9 | measiuns 31480 | . . 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 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ∪ ran sigAlgebra) |
34 | nfv 1914 | . . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑 | |
35 | nfcv 2980 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘 | |
36 | 35 | nfel1 2997 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑘 ∈ ℕ |
37 | 27 | nfel1 2997 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆 |
38 | 36, 37 | nfim 1896 | . . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
39 | 34, 38 | nfim 1896 | . . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
40 | eleq1w 2898 | . . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ)) | |
41 | 28 | eleq1d 2900 | . . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐵 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
42 | 40, 41 | imbi12d 347 | . . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑛 ∈ ℕ → 𝐵 ∈ 𝑆) ↔ (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆))) |
43 | 42 | imbi2d 343 | . . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝜑 → (𝑛 ∈ ℕ → 𝐵 ∈ 𝑆)) ↔ (𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)))) |
44 | 9 | ex 415 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ → 𝐵 ∈ 𝑆)) |
45 | 39, 43, 44 | chvarfv 2241 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
46 | 45 | ralrimiv 3184 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
47 | fzossnn 13089 | . . . . . . . . . 10 ⊢ (1..^𝑛) ⊆ ℕ | |
48 | ssralv 4036 | . . . . . . . . . 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 31384 | . . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) | |
51 | 49, 50 | sylan2 594 | . . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
52 | 8, 46, 51 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
53 | 52 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
54 | difelsiga 31396 | . . . . . 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 1367 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ∈ 𝑆) |
56 | measvxrge0 31468 | . . . . 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 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ∈ (0[,]+∞)) |
58 | difssd 4112 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ⊆ 𝐵) | |
59 | 17, 55, 9, 58 | measssd 31478 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ (𝑀‘𝐵)) |
60 | 32, 57, 19, 59 | esumle 31321 | . . 3 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
61 | 31, 60 | eqbrtrd 5091 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
62 | 6, 15, 24, 26, 61 | xrletrd 12558 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 Vcvv 3497 ⦋csb 3886 ∖ cdif 3936 ⊆ wss 3939 ∪ cuni 4841 ∪ ciun 4922 class class class wbr 5069 ran crn 5559 ‘cfv 6358 (class class class)co 7159 0cc0 10540 1c1 10541 +∞cpnf 10675 ℝ*cxr 10677 ≤ cle 10679 ℕcn 11641 [,]cicc 12744 ..^cfzo 13036 Σ*cesum 31290 sigAlgebracsiga 31371 measurescmeas 31458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-ac2 9888 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-disj 5035 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 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 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-dju 9333 df-card 9371 df-acn 9374 df-ac 9545 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-sum 15046 df-ef 15424 df-sin 15426 df-cos 15427 df-pi 15429 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-ordt 16777 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-ps 17813 df-tsr 17814 df-plusf 17854 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-mulg 18228 df-subg 18279 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-subrg 19536 df-abv 19591 df-lmod 19639 df-scaf 19640 df-sra 19947 df-rgmod 19948 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-lp 21747 df-perf 21748 df-cn 21838 df-cnp 21839 df-haus 21926 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-tmd 22683 df-tgp 22684 df-tsms 22738 df-trg 22771 df-xms 22933 df-ms 22934 df-tms 22935 df-nm 23195 df-ngp 23196 df-nrg 23198 df-nlm 23199 df-ii 23488 df-cncf 23489 df-limc 24467 df-dv 24468 df-log 25143 df-esum 31291 df-siga 31372 df-meas 31459 |
This theorem is referenced by: (None) |
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