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Theorem measunl 34206

Description: A measure is sub-additive with respect to union. (Contributed by Thierry Arnoux, 20-Oct-2017.)

**Hypotheses**
Ref	Expression
measunl.1	⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
measunl.2	⊢ (𝜑 → 𝐴 ∈ 𝑆)
measunl.3	⊢ (𝜑 → 𝐵 ∈ 𝑆)

**Assertion**
Ref	Expression
measunl	⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +_𝑒 (𝑀‘𝐵)))

**Proof of Theorem measunl**
Step	Hyp	Ref	Expression
1		undif1 4439	. . . 4 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
2	1	fveq2i 6861	. . 3 ⊢ (𝑀‘((𝐴 ∖ 𝐵) ∪ 𝐵)) = (𝑀‘(𝐴 ∪ 𝐵))
3		measunl.1	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
4		measbase 34187	. . . . . 6 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
6		measunl.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
7		measunl.3	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
8		difelsiga 34123	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)
9	5, 6, 7, 8	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝑆)
10		disjdifr 4436	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐵) = ∅
11	10	a1i 11	. . . 4 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∩ 𝐵) = ∅)
12		measun 34201	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ((𝐴 ∖ 𝐵) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ ((𝐴 ∖ 𝐵) ∩ 𝐵) = ∅) → (𝑀‘((𝐴 ∖ 𝐵) ∪ 𝐵)) = ((𝑀‘(𝐴 ∖ 𝐵)) +_𝑒 (𝑀‘𝐵)))
13	3, 9, 7, 11, 12	syl121anc 1377	. . 3 ⊢ (𝜑 → (𝑀‘((𝐴 ∖ 𝐵) ∪ 𝐵)) = ((𝑀‘(𝐴 ∖ 𝐵)) +_𝑒 (𝑀‘𝐵)))
14	2, 13	eqtr3id 2778	. 2 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘(𝐴 ∖ 𝐵)) +_𝑒 (𝑀‘𝐵)))
15		iccssxr 13391	. . . 4 ⊢ (0[,]+∞) ⊆ ℝ^*
16		measvxrge0 34195	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → (𝑀‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞))
17	3, 9, 16	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞))
18	15, 17	sselid 3944	. . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ∈ ℝ^*)
19		measvxrge0 34195	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞))
20	3, 6, 19	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞))
21	15, 20	sselid 3944	. . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ^*)
22		measvxrge0 34195	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ (0[,]+∞))
23	3, 7, 22	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ∈ (0[,]+∞))
24	15, 23	sselid 3944	. . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ^*)
25		inelsiga 34125	. . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)
26	5, 6, 7, 25	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑆)
27		measvxrge0 34195	. . . . . . . 8 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∩ 𝐵) ∈ 𝑆) → (𝑀‘(𝐴 ∩ 𝐵)) ∈ (0[,]+∞))
28	3, 26, 27	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝐴 ∩ 𝐵)) ∈ (0[,]+∞))
29		elxrge0 13418	. . . . . . 7 ⊢ ((𝑀‘(𝐴 ∩ 𝐵)) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐴 ∩ 𝐵)) ∈ ℝ^* ∧ 0 ≤ (𝑀‘(𝐴 ∩ 𝐵))))
30	28, 29	sylib 218	. . . . . 6 ⊢ (𝜑 → ((𝑀‘(𝐴 ∩ 𝐵)) ∈ ℝ^* ∧ 0 ≤ (𝑀‘(𝐴 ∩ 𝐵))))
31	30	simprd 495	. . . . 5 ⊢ (𝜑 → 0 ≤ (𝑀‘(𝐴 ∩ 𝐵)))
32	15, 28	sselid 3944	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐴 ∩ 𝐵)) ∈ ℝ^*)
33		xraddge02 32680	. . . . . 6 ⊢ (((𝑀‘(𝐴 ∖ 𝐵)) ∈ ℝ^* ∧ (𝑀‘(𝐴 ∩ 𝐵)) ∈ ℝ^*) → (0 ≤ (𝑀‘(𝐴 ∩ 𝐵)) → (𝑀‘(𝐴 ∖ 𝐵)) ≤ ((𝑀‘(𝐴 ∖ 𝐵)) +_𝑒 (𝑀‘(𝐴 ∩ 𝐵)))))
34	18, 32, 33	syl2anc 584	. . . . 5 ⊢ (𝜑 → (0 ≤ (𝑀‘(𝐴 ∩ 𝐵)) → (𝑀‘(𝐴 ∖ 𝐵)) ≤ ((𝑀‘(𝐴 ∖ 𝐵)) +_𝑒 (𝑀‘(𝐴 ∩ 𝐵)))))
35	31, 34	mpd 15	. . . 4 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ≤ ((𝑀‘(𝐴 ∖ 𝐵)) +_𝑒 (𝑀‘(𝐴 ∩ 𝐵))))
36		uncom 4121	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐵))
37		inundif 4442	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴
38	36, 37	eqtr3i 2754	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐵)) = 𝐴
39	38	fveq2i 6861	. . . . 5 ⊢ (𝑀‘((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐵))) = (𝑀‘𝐴)
40		incom 4172	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∩ 𝐵))
41		inindif 4338	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) = ∅
42	40, 41	eqtr3i 2754	. . . . . . 7 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐴 ∩ 𝐵)) = ∅
43	42	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∩ (𝐴 ∩ 𝐵)) = ∅)
44		measun 34201	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ((𝐴 ∖ 𝐵) ∈ 𝑆 ∧ (𝐴 ∩ 𝐵) ∈ 𝑆) ∧ ((𝐴 ∖ 𝐵) ∩ (𝐴 ∩ 𝐵)) = ∅) → (𝑀‘((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐵))) = ((𝑀‘(𝐴 ∖ 𝐵)) +_𝑒 (𝑀‘(𝐴 ∩ 𝐵))))
45	3, 9, 26, 43, 44	syl121anc 1377	. . . . 5 ⊢ (𝜑 → (𝑀‘((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐵))) = ((𝑀‘(𝐴 ∖ 𝐵)) +_𝑒 (𝑀‘(𝐴 ∩ 𝐵))))
46	39, 45	eqtr3id 2778	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) = ((𝑀‘(𝐴 ∖ 𝐵)) +_𝑒 (𝑀‘(𝐴 ∩ 𝐵))))
47	35, 46	breqtrrd 5135	. . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ≤ (𝑀‘𝐴))
48		xleadd1a 13213	. . 3 ⊢ ((((𝑀‘(𝐴 ∖ 𝐵)) ∈ ℝ^* ∧ (𝑀‘𝐴) ∈ ℝ^* ∧ (𝑀‘𝐵) ∈ ℝ^*) ∧ (𝑀‘(𝐴 ∖ 𝐵)) ≤ (𝑀‘𝐴)) → ((𝑀‘(𝐴 ∖ 𝐵)) +_𝑒 (𝑀‘𝐵)) ≤ ((𝑀‘𝐴) +_𝑒 (𝑀‘𝐵)))
49	18, 21, 24, 47, 48	syl31anc 1375	. 2 ⊢ (𝜑 → ((𝑀‘(𝐴 ∖ 𝐵)) +_𝑒 (𝑀‘𝐵)) ≤ ((𝑀‘𝐴) +_𝑒 (𝑀‘𝐵)))
50	14, 49	eqbrtrd 5129	1 ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +_𝑒 (𝑀‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ∅c0 4296 ∪ cuni 4871 class class class wbr 5107 ran crn 5639 ‘cfv 6511 (class class class)co 7387 0cc0 11068 +∞cpnf 11205 ℝ^*cxr 11207 ≤ cle 11209 +_𝑒 cxad 13070 [,]cicc 13309 sigAlgebracsiga 34098 measurescmeas 34185

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-ac2 10416 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-acn 9895 df-ac 10069 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-ordt 17464 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-ps 18525 df-tsr 18526 df-plusf 18566 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-subrng 20455 df-subrg 20479 df-abv 20718 df-lmod 20768 df-scaf 20769 df-sra 21080 df-rgmod 21081 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-tmd 23959 df-tgp 23960 df-tsms 24014 df-trg 24047 df-xms 24208 df-ms 24209 df-tms 24210 df-nm 24470 df-ngp 24471 df-nrg 24473 df-nlm 24474 df-ii 24770 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-esum 34018 df-siga 34099 df-meas 34186

This theorem is referenced by: aean 34234

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