measiun - Mathbox for Thierry Arnoux

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Theorem measiun 34182

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 13490	. . 3 ⊢ (0[,]+∞) ⊆ ℝ^*
2		measiun.1	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
3		measiun.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
4		measvxrge0 34169	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞))
5	2, 3, 4	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞))
6	1, 5	sselid 4006	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ^*)
7		measbase 34161	. . . . . 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 3152	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐵 ∈ 𝑆)
11		sigaclcu2 34084	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑛 ∈ ℕ 𝐵 ∈ 𝑆) → ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆)
12	8, 10, 11	syl2anc 583	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆)
13		measvxrge0 34169	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∪ 𝑛 ∈ ℕ 𝐵 ∈ 𝑆) → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ (0[,]+∞))
14	2, 12, 13	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ (0[,]+∞))
15	1, 14	sselid 4006	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ∈ ℝ^*)
16		nnex 12299	. . . 4 ⊢ ℕ ∈ V
17	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆))
18		measvxrge0 34169	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ (0[,]+∞))
19	17, 9, 18	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝐵) ∈ (0[,]+∞))
20	19	ralrimiva 3152	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑀‘𝐵) ∈ (0[,]+∞))
21		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑛ℕ
22	21	esumcl 33994	. . . 4 ⊢ ((ℕ ∈ V ∧ ∀𝑛 ∈ ℕ (𝑀‘𝐵) ∈ (0[,]+∞)) → Σ^*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ (0[,]+∞))
23	16, 20, 22	sylancr 586	. . 3 ⊢ (𝜑 → Σ^*𝑛 ∈ ℕ(𝑀‘𝐵) ∈ (0[,]+∞))
24	1, 23	sselid 4006	. 2 ⊢ (𝜑 → Σ^𝑛 ∈ ℕ(𝑀‘𝐵) ∈ ℝ^)
25		measiun.4	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐵)
26	2, 3, 12, 25	measssd 34179	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘∪ 𝑛 ∈ ℕ 𝐵))
27		nfcsb1v 3946	. . . 4 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵
28		csbeq1a 3935	. . . 4 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
29		eqidd 2741	. . . . 5 ⊢ (𝜑 → ℕ = ℕ)
30	29	orcd 872	. . . 4 ⊢ (𝜑 → (ℕ = ℕ ∨ ℕ = (1..^𝑚)))
31	27, 28, 30, 2, 9	measiuns 34181	. . 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 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝜑
35		nfcv 2908	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘
36	35	nfel1 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑘 ∈ ℕ
37	27	nfel1 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆
38	36, 37	nfim 1895	. . . . . . . . . . 11 ⊢ Ⅎ𝑛(𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)
39	34, 38	nfim 1895	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆))
40		eleq1w 2827	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℕ ↔ 𝑘 ∈ ℕ))
41	28	eleq1d 2829	. . . . . . . . . . . 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 412	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ → 𝐵 ∈ 𝑆))
45	39, 43, 44	chvarfv 2241	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆))
46	45	ralrimiv 3151	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ⦋𝑘 / 𝑛⦌𝐵 ∈ 𝑆)
47		fzossnn 13765	. . . . . . . . . 10 ⊢ (1..^𝑛) ⊆ ℕ
48		ssralv 4077	. . . . . . . . . 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 34085	. . . . . . . . 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 583	. . . . . . 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 34097	. . . . . 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 1371	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ∈ 𝑆)
56		measvxrge0 34169	. . . . 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 583	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ∈ (0[,]+∞))
58		difssd 4160	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵) ⊆ 𝐵)
59	17, 55, 9, 58	measssd 34179	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ (𝑀‘𝐵))
60	32, 57, 19, 59	esumle 34022	. . 3 ⊢ (𝜑 → Σ^𝑛 ∈ ℕ(𝑀‘(𝐵 ∖ ∪ 𝑘 ∈ (1..^𝑛)⦋𝑘 / 𝑛⦌𝐵)) ≤ Σ^𝑛 ∈ ℕ(𝑀‘𝐵))
61	31, 60	eqbrtrd 5188	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ 𝐵) ≤ Σ^*𝑛 ∈ ℕ(𝑀‘𝐵))
62	6, 15, 24, 26, 61	xrletrd 13224	1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ Σ^*𝑛 ∈ ℕ(𝑀‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ⦋csb 3921 ∖ cdif 3973 ⊆ wss 3976 ∪ cuni 4931 ∪ ciun 5015 class class class wbr 5166 ran crn 5701 ‘cfv 6573 (class class class)co 7448 0cc0 11184 1c1 11185 +∞cpnf 11321 ℝ^*cxr 11323 ≤ cle 11325 ℕcn 12293 [,]cicc 13410 ..^cfzo 13711 Σ^*cesum 33991 sigAlgebracsiga 34072 measurescmeas 34159

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-ac2 10532 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-ac 10185 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-ordt 17561 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-ps 18636 df-tsr 18637 df-plusf 18677 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-subrng 20572 df-subrg 20597 df-abv 20832 df-lmod 20882 df-scaf 20883 df-sra 21195 df-rgmod 21196 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-tmd 24101 df-tgp 24102 df-tsms 24156 df-trg 24189 df-xms 24351 df-ms 24352 df-tms 24353 df-nm 24616 df-ngp 24617 df-nrg 24619 df-nlm 24620 df-ii 24922 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-esum 33992 df-siga 34073 df-meas 34160

This theorem is referenced by: (None)

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