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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 12669 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | measiun.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) | |
3 | measiun.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | measvxrge0 31081 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | |
5 | 2, 3, 4 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
6 | 1, 5 | sseldi 3887 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
7 | measbase 31073 | . . . . . 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 3149 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐵 ∈ 𝑆) |
11 | sigaclcu2 30996 | . . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑛 ∈ ℕ 𝐵 ∈ 𝑆) → ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) | |
12 | 8, 10, 11 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) |
13 | measvxrge0 31081 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ (0[,]+∞)) | |
14 | 2, 12, 13 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ (0[,]+∞)) |
15 | 1, 14 | sseldi 3887 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ ℝ*) |
16 | nnex 11492 | . . . 4 ⊢ ℕ ∈ V | |
17 | 2 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆)) |
18 | measvxrge0 31081 | . . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ (0[,]+∞)) | |
19 | 17, 9, 18 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝐵) ∈ (0[,]+∞)) |
20 | 19 | ralrimiva 3149 | . . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑀‘𝐵) ∈ (0[,]+∞)) |
21 | nfcv 2949 | . . . . 5 ⊢ Ⅎ𝑛ℕ | |
22 | 21 | esumcl 30906 | . . . 4 ⊢ ((ℕ ∈ V ∧ ∀𝑛 ∈ ℕ (𝑀‘𝐵) ∈ (0[,]+∞)) → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ (0[,]+∞)) |
23 | 16, 20, 22 | sylancr 587 | . . 3 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ (0[,]+∞)) |
24 | 1, 23 | sseldi 3887 | . 2 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ ℝ*) |
25 | measiun.4 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐵) | |
26 | 2, 3, 12, 25 | measssd 31091 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘∪ 𝑛 ∈ ℕ 𝐵)) |
27 | nfcsb1v 3833 | . . . 4 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 | |
28 | csbeq1a 3824 | . . . 4 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵) | |
29 | eqidd 2796 | . . . . 5 ⊢ (𝜑 → ℕ = ℕ) | |
30 | 29 | orcd 870 | . . . 4 ⊢ (𝜑 → (ℕ = ℕ ∨ ℕ = (1..^𝑚))) |
31 | 27, 28, 30, 2, 9 | measiuns 31093 | . . 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 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ∪ ran sigAlgebra) |
34 | nfv 1892 | . . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑 | |
35 | nfcv 2949 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘 | |
36 | 35 | nfel1 2963 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑘 ∈ ℕ |
37 | 27 | nfel1 2963 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆 |
38 | 36, 37 | nfim 1878 | . . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
39 | 34, 38 | nfim 1878 | . . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
40 | eleq1w 2865 | . . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ)) | |
41 | 28 | eleq1d 2867 | . . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐵 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
42 | 40, 41 | imbi12d 346 | . . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑛 ∈ ℕ → 𝐵 ∈ 𝑆) ↔ (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆))) |
43 | 42 | imbi2d 342 | . . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝜑 → (𝑛 ∈ ℕ → 𝐵 ∈ 𝑆)) ↔ (𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)))) |
44 | 9 | ex 413 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ → 𝐵 ∈ 𝑆)) |
45 | 39, 43, 44 | chvar 2369 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
46 | 45 | ralrimiv 3148 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
47 | fzossnn 12936 | . . . . . . . . . 10 ⊢ (1..^𝑛) ⊆ ℕ | |
48 | ssralv 3954 | . . . . . . . . . 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 30997 | . . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) | |
51 | 49, 50 | sylan2 592 | . . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
52 | 8, 46, 51 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
53 | 52 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
54 | difelsiga 31009 | . . . . . 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 1364 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ∈ 𝑆) |
56 | measvxrge0 31081 | . . . . 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 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ∈ (0[,]+∞)) |
58 | difssd 4030 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ⊆ 𝐵) | |
59 | 17, 55, 9, 58 | measssd 31091 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ (𝑀‘𝐵)) |
60 | 32, 57, 19, 59 | esumle 30934 | . . 3 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
61 | 31, 60 | eqbrtrd 4984 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
62 | 6, 15, 24, 26, 61 | xrletrd 12405 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∀wral 3105 Vcvv 3437 ⦋csb 3811 ∖ cdif 3856 ⊆ wss 3859 ∪ cuni 4745 ∪ ciun 4825 class class class wbr 4962 ran crn 5444 ‘cfv 6225 (class class class)co 7016 0cc0 10383 1c1 10384 +∞cpnf 10518 ℝ*cxr 10520 ≤ cle 10522 ℕcn 11486 [,]cicc 12591 ..^cfzo 12883 Σ*cesum 30903 sigAlgebracsiga 30984 measurescmeas 31071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-ac2 9731 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-disj 4931 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-pm 8259 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-fi 8721 df-sup 8752 df-inf 8753 df-oi 8820 df-dju 9176 df-card 9214 df-acn 9217 df-ac 9388 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-ioc 12593 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-fl 13012 df-mod 13088 df-seq 13220 df-exp 13280 df-fac 13484 df-bc 13513 df-hash 13541 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 df-sin 15256 df-cos 15257 df-pi 15259 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-ordt 16603 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-ps 17639 df-tsr 17640 df-plusf 17680 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-mhm 17774 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-mulg 17982 df-subg 18030 df-cntz 18188 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-cring 18990 df-subrg 19223 df-abv 19278 df-lmod 19326 df-scaf 19327 df-sra 19634 df-rgmod 19635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-ntr 21312 df-cls 21313 df-nei 21390 df-lp 21428 df-perf 21429 df-cn 21519 df-cnp 21520 df-haus 21607 df-tx 21854 df-hmeo 22047 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-tmd 22364 df-tgp 22365 df-tsms 22418 df-trg 22451 df-xms 22613 df-ms 22614 df-tms 22615 df-nm 22875 df-ngp 22876 df-nrg 22878 df-nlm 22879 df-ii 23168 df-cncf 23169 df-limc 24147 df-dv 24148 df-log 24821 df-esum 30904 df-siga 30985 df-meas 31072 |
This theorem is referenced by: (None) |
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