measiun - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > measiun		Structured version Visualization version GIF version

Theorem measiun 32292

Description: A measure is sub-additive. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
measiun	⊢ (𝜑 → (𝑀‘𝐴) ≤ Σ^*𝑛 ∈ ℕ(𝑀‘𝐵))

Distinct variable groups: 𝜑,𝑛 𝑆,𝑛 𝑛,𝑀 Allowed substitution hints: 𝐴(𝑛) 𝐵(𝑛)

Proof of Theorem measiun Dummy variables 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 13232	. . 3 ⊢ (0[,]+∞) ⊆ ℝ^*
2		measiun.1	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
3		measiun.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
4		measvxrge0 32279	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞))
5	2, 3, 4	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞))
6	1, 5	sselid 3928	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ^*)
7		measbase 32271	. . . . . 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 3140	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐵 ∈ 𝑆)
11		sigaclcu2 32194	. . . . 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 32279	. . . 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	sselid 3928	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ ℝ^*)
16		nnex 12049	. . . 4 ⊢ ℕ ∈ V
17	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆))
18		measvxrge0 32279	. . . . . 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 3140	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑀‘𝐵) ∈ (0[,]+∞))
21		nfcv 2905	. . . . 5 ⊢ Ⅎ𝑛ℕ
22	21	esumcl 32104	. . . 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	sselid 3928	. 2 ⊢ (𝜑 → Σ^𝑛 ∈ ℕ(𝑀‘𝐵) ∈ ℝ^)
25		measiun.4	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐵)
26	2, 3, 12, 25	measssd 32289	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘∪ 𝑛 ∈ ℕ 𝐵))
27		nfcsb1v 3866	. . . 4 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵
28		csbeq1a 3855	. . . 4 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
29		eqidd 2738	. . . . 5 ⊢ (𝜑 → ℕ = ℕ)
30	29	orcd 870	. . . 4 ⊢ (𝜑 → (ℕ = ℕ ∨ ℕ = (1..^𝑚)))
31	27, 28, 30, 2, 9	measiuns 32291	. . 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 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑
35		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘
36	35	nfel1 2921	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑘 ∈ ℕ
37	27	nfel1 2921	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆
38	36, 37	nfim 1898	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)
39	34, 38	nfim 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆))
40		eleq1w 2820	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ))
41	28	eleq1d 2822	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐵 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆))
42	40, 41	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → ((𝑛 ∈ ℕ → 𝐵 ∈ 𝑆) ↔ (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)))
43	42	imbi2d 340	. . . . . . . . . 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	chvarfv 2232	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆))
46	45	ralrimiv 3139	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)
47		fzossnn 13506	. . . . . . . . . 10 ⊢ (1..^𝑛) ⊆ ℕ
48		ssralv 3996	. . . . . . . . . 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 32195	. . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)
51	49, 50	sylan2 593	. . . . . . . 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 32207	. . . . . 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 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ∈ 𝑆)
56		measvxrge0 32279	. . . . 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 4077	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ⊆ 𝐵)
59	17, 55, 9, 58	measssd 32289	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ (𝑀‘𝐵))
60	32, 57, 19, 59	esumle 32132	. . 3 ⊢ (𝜑 → Σ^𝑛 ∈ ℕ(𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ Σ^𝑛 ∈ ℕ(𝑀‘𝐵))
61	31, 60	eqbrtrd 5107	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ≤ Σ^*𝑛 ∈ ℕ(𝑀‘𝐵))
62	6, 15, 24, 26, 61	xrletrd 12966	1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ Σ^*𝑛 ∈ ℕ(𝑀‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3062 Vcvv 3441 ⦋csb 3841 ∖ cdif 3893 ⊆ wss 3896 ∪ cuni 4848 ∪ ciun 4935 class class class wbr 5085 ran crn 5606 ‘cfv 6463 (class class class)co 7313 0cc0 10941 1c1 10942 +∞cpnf 11076 ℝ^*cxr 11078 ≤ cle 11080 ℕcn 12043 [,]cicc 13152 ..^cfzo 13452 Σ^*cesum 32101 sigAlgebracsiga 32182 measurescmeas 32269

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5222 ax-sep 5236 ax-nul 5243 ax-pow 5301 ax-pr 5365 ax-un 7626 ax-inf2 9467 ax-ac2 10289 ax-cnex 10997 ax-resscn 10998 ax-1cn 10999 ax-icn 11000 ax-addcl 11001 ax-addrcl 11002 ax-mulcl 11003 ax-mulrcl 11004 ax-mulcom 11005 ax-addass 11006 ax-mulass 11007 ax-distr 11008 ax-i2m1 11009 ax-1ne0 11010 ax-1rid 11011 ax-rnegex 11012 ax-rrecex 11013 ax-cnre 11014 ax-pre-lttri 11015 ax-pre-lttrn 11016 ax-pre-ltadd 11017 ax-pre-mulgt0 11018 ax-pre-sup 11019 ax-addf 11020 ax-mulf 11021

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4470 df-pw 4545 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4849 df-int 4891 df-iun 4937 df-iin 4938 df-disj 5051 df-br 5086 df-opab 5148 df-mpt 5169 df-tr 5203 df-id 5505 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5560 df-se 5561 df-we 5562 df-xp 5611 df-rel 5612 df-cnv 5613 df-co 5614 df-dm 5615 df-rn 5616 df-res 5617 df-ima 5618 df-pred 6222 df-ord 6289 df-on 6290 df-lim 6291 df-suc 6292 df-iota 6415 df-fun 6465 df-fn 6466 df-f 6467 df-f1 6468 df-fo 6469 df-f1o 6470 df-fv 6471 df-isom 6472 df-riota 7270 df-ov 7316 df-oprab 7317 df-mpo 7318 df-of 7571 df-om 7756 df-1st 7874 df-2nd 7875 df-supp 8023 df-frecs 8142 df-wrecs 8173 df-recs 8247 df-rdg 8286 df-1o 8342 df-2o 8343 df-er 8544 df-map 8663 df-pm 8664 df-ixp 8732 df-en 8780 df-dom 8781 df-sdom 8782 df-fin 8783 df-fsupp 9197 df-fi 9238 df-sup 9269 df-inf 9270 df-oi 9337 df-dju 9727 df-card 9765 df-acn 9768 df-ac 9942 df-pnf 11081 df-mnf 11082 df-xr 11083 df-ltxr 11084 df-le 11085 df-sub 11277 df-neg 11278 df-div 11703 df-nn 12044 df-2 12106 df-3 12107 df-4 12108 df-5 12109 df-6 12110 df-7 12111 df-8 12112 df-9 12113 df-n0 12304 df-z 12390 df-dec 12508 df-uz 12653 df-q 12759 df-rp 12801 df-xneg 12918 df-xadd 12919 df-xmul 12920 df-ioo 13153 df-ioc 13154 df-ico 13155 df-icc 13156 df-fz 13310 df-fzo 13453 df-fl 13582 df-mod 13660 df-seq 13792 df-exp 13853 df-fac 14058 df-bc 14087 df-hash 14115 df-shft 14847 df-cj 14879 df-re 14880 df-im 14881 df-sqrt 15015 df-abs 15016 df-limsup 15249 df-clim 15266 df-rlim 15267 df-sum 15467 df-ef 15846 df-sin 15848 df-cos 15849 df-pi 15851 df-struct 16915 df-sets 16932 df-slot 16950 df-ndx 16962 df-base 16980 df-ress 17009 df-plusg 17042 df-mulr 17043 df-starv 17044 df-sca 17045 df-vsca 17046 df-ip 17047 df-tset 17048 df-ple 17049 df-ds 17051 df-unif 17052 df-hom 17053 df-cco 17054 df-rest 17200 df-topn 17201 df-0g 17219 df-gsum 17220 df-topgen 17221 df-pt 17222 df-prds 17225 df-ordt 17279 df-xrs 17280 df-qtop 17285 df-imas 17286 df-xps 17288 df-mre 17362 df-mrc 17363 df-acs 17365 df-ps 18351 df-tsr 18352 df-plusf 18392 df-mgm 18393 df-sgrp 18442 df-mnd 18453 df-mhm 18497 df-submnd 18498 df-grp 18647 df-minusg 18648 df-sbg 18649 df-mulg 18768 df-subg 18819 df-cntz 18990 df-cmn 19455 df-abl 19456 df-mgp 19788 df-ur 19805 df-ring 19852 df-cring 19853 df-subrg 20093 df-abv 20148 df-lmod 20196 df-scaf 20197 df-sra 20505 df-rgmod 20506 df-psmet 20660 df-xmet 20661 df-met 20662 df-bl 20663 df-mopn 20664 df-fbas 20665 df-fg 20666 df-cnfld 20669 df-top 22114 df-topon 22131 df-topsp 22153 df-bases 22167 df-cld 22241 df-ntr 22242 df-cls 22243 df-nei 22320 df-lp 22358 df-perf 22359 df-cn 22449 df-cnp 22450 df-haus 22537 df-tx 22784 df-hmeo 22977 df-fil 23068 df-fm 23160 df-flim 23161 df-flf 23162 df-tmd 23294 df-tgp 23295 df-tsms 23349 df-trg 23382 df-xms 23544 df-ms 23545 df-tms 23546 df-nm 23809 df-ngp 23810 df-nrg 23812 df-nlm 23813 df-ii 24111 df-cncf 24112 df-limc 25101 df-dv 25102 df-log 25783 df-esum 32102 df-siga 32183 df-meas 32270

This theorem is referenced by: (None)

W3C validator