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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 12127 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | measiun.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) | |
3 | measiun.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | measvxrge0 29595 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | |
5 | 2, 3, 4 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞)) |
6 | 1, 5 | sseldi 3566 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*) |
7 | measbase 29587 | . . . . . 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 2949 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐵 ∈ 𝑆) |
11 | sigaclcu2 29510 | . . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑛 ∈ ℕ 𝐵 ∈ 𝑆) → ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) | |
12 | 8, 10, 11 | syl2anc 691 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) |
13 | measvxrge0 29595 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ (0[,]+∞)) | |
14 | 2, 12, 13 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ (0[,]+∞)) |
15 | 1, 14 | sseldi 3566 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ ℝ*) |
16 | nnex 10903 | . . . 4 ⊢ ℕ ∈ V | |
17 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆)) |
18 | measvxrge0 29595 | . . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ (0[,]+∞)) | |
19 | 17, 9, 18 | syl2anc 691 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝐵) ∈ (0[,]+∞)) |
20 | 19 | ralrimiva 2949 | . . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑀‘𝐵) ∈ (0[,]+∞)) |
21 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑛ℕ | |
22 | 21 | esumcl 29419 | . . . 4 ⊢ ((ℕ ∈ V ∧ ∀𝑛 ∈ ℕ (𝑀‘𝐵) ∈ (0[,]+∞)) → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ (0[,]+∞)) |
23 | 16, 20, 22 | sylancr 694 | . . 3 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ (0[,]+∞)) |
24 | 1, 23 | sseldi 3566 | . 2 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ ℝ*) |
25 | measiun.4 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐵) | |
26 | 2, 3, 12, 25 | measssd 29605 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘∪ 𝑛 ∈ ℕ 𝐵)) |
27 | nfcsb1v 3515 | . . . 4 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 | |
28 | csbeq1a 3508 | . . . 4 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵) | |
29 | eqidd 2611 | . . . . 5 ⊢ (𝜑 → ℕ = ℕ) | |
30 | 29 | orcd 406 | . . . 4 ⊢ (𝜑 → (ℕ = ℕ ∨ ℕ = (1..^𝑚))) |
31 | 27, 28, 30, 2, 9 | measiuns 29607 | . . 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 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ∪ ran sigAlgebra) |
34 | nfv 1830 | . . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑 | |
35 | nfcv 2751 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘 | |
36 | 35 | nfel1 2765 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑘 ∈ ℕ |
37 | 27 | nfel1 2765 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆 |
38 | 36, 37 | nfim 1813 | . . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
39 | 34, 38 | nfim 1813 | . . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
40 | eleq1 2676 | . . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ)) | |
41 | 28 | eleq1d 2672 | . . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐵 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
42 | 40, 41 | imbi12d 333 | . . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑛 ∈ ℕ → 𝐵 ∈ 𝑆) ↔ (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆))) |
43 | 42 | imbi2d 329 | . . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝜑 → (𝑛 ∈ ℕ → 𝐵 ∈ 𝑆)) ↔ (𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)))) |
44 | 9 | ex 449 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ → 𝐵 ∈ 𝑆)) |
45 | 39, 43, 44 | chvar 2250 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)) |
46 | 45 | ralrimiv 2948 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
47 | fzossnn 12384 | . . . . . . . . . 10 ⊢ (1..^𝑛) ⊆ ℕ | |
48 | ssralv 3629 | . . . . . . . . . 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 29511 | . . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) | |
51 | 49, 50 | sylan2 490 | . . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
52 | 8, 46, 51 | syl2anc 691 | . . . . . . 7 ⊢ (𝜑 → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
53 | 52 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) |
54 | difelsiga 29523 | . . . . . 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 1318 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ∈ 𝑆) |
56 | measvxrge0 29595 | . . . . 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 691 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ∈ (0[,]+∞)) |
58 | difssd 3700 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ⊆ 𝐵) | |
59 | 17, 55, 9, 58 | measssd 29605 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ (𝑀‘𝐵)) |
60 | 32, 57, 19, 59 | esumle 29447 | . . 3 ⊢ (𝜑 → Σ*𝑛 ∈ ℕ(𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
61 | 31, 60 | eqbrtrd 4605 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
62 | 6, 15, 24, 26, 61 | xrletrd 11869 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ Σ*𝑛 ∈ ℕ(𝑀‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⦋csb 3499 ∖ cdif 3537 ⊆ wss 3540 ∪ cuni 4372 ∪ ciun 4455 class class class wbr 4583 ran crn 5039 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 +∞cpnf 9950 ℝ*cxr 9952 ≤ cle 9954 ℕcn 10897 [,]cicc 12049 ..^cfzo 12334 Σ*cesum 29416 sigAlgebracsiga 29497 measurescmeas 29585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-ac2 9168 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-ac 8822 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-ordt 15984 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-ps 17023 df-tsr 17024 df-plusf 17064 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-abv 18640 df-lmod 18688 df-scaf 18689 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tmd 21686 df-tgp 21687 df-tsms 21740 df-trg 21773 df-xms 21935 df-ms 21936 df-tms 21937 df-nm 22197 df-ngp 22198 df-nrg 22200 df-nlm 22201 df-ii 22488 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-esum 29417 df-siga 29498 df-meas 29586 |
This theorem is referenced by: (None) |
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