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Theorem measssd 34399

Description: A measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 28-Dec-2016.)

**Hypotheses**
Ref	Expression
measssd.1	⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
measssd.2	⊢ (𝜑 → 𝐴 ∈ 𝑆)
measssd.3	⊢ (𝜑 → 𝐵 ∈ 𝑆)
measssd.4	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

**Assertion**
Ref	Expression
measssd	⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))

Proof of Theorem measssd Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		measssd.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
2		measbase 34381	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
4		measssd.3	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
5		measssd.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
6		difelsiga 34317	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆)
7	3, 4, 5, 6	syl3anc 1379	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆)
8		measvxrge0 34389	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞))
9	1, 7, 8	syl2anc 590	. . . 4 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞))
10		elxrge0 13401	. . . . 5 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ^* ∧ 0 ≤ (𝑀‘(𝐵 ∖ 𝐴))))
11	10	simprbi 498	. . . 4 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) → 0 ≤ (𝑀‘(𝐵 ∖ 𝐴)))
12	9, 11	syl 17	. . 3 ⊢ (𝜑 → 0 ≤ (𝑀‘(𝐵 ∖ 𝐴)))
13		measvxrge0 34389	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞))
14	1, 5, 13	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞))
15		elxrge0 13401	. . . . . 6 ⊢ ((𝑀‘𝐴) ∈ (0[,]+∞) ↔ ((𝑀‘𝐴) ∈ ℝ^* ∧ 0 ≤ (𝑀‘𝐴)))
16	15	simplbi 497	. . . . 5 ⊢ ((𝑀‘𝐴) ∈ (0[,]+∞) → (𝑀‘𝐴) ∈ ℝ^*)
17	14, 16	syl 17	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ^*)
18	10	simplbi 497	. . . . 5 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ^*)
19	9, 18	syl 17	. . . 4 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ^*)
20		xraddge02 32849	. . . 4 ⊢ (((𝑀‘𝐴) ∈ ℝ^* ∧ (𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ^*) → (0 ≤ (𝑀‘(𝐵 ∖ 𝐴)) → (𝑀‘𝐴) ≤ ((𝑀‘𝐴) +_𝑒 (𝑀‘(𝐵 ∖ 𝐴)))))
21	17, 19, 20	syl2anc 590	. . 3 ⊢ (𝜑 → (0 ≤ (𝑀‘(𝐵 ∖ 𝐴)) → (𝑀‘𝐴) ≤ ((𝑀‘𝐴) +_𝑒 (𝑀‘(𝐵 ∖ 𝐴)))))
22	12, 21	mpd 15	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ≤ ((𝑀‘𝐴) +_𝑒 (𝑀‘(𝐵 ∖ 𝐴))))
23		prssi 4752	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆)
24	5, 7, 23	syl2anc 590	. . . . 5 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆)
25		prex 5367	. . . . . 6 ⊢ {𝐴, (𝐵 ∖ 𝐴)} ∈ V
26	25	elpw 4533	. . . . 5 ⊢ ({𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆 ↔ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆)
27	24, 26	sylibr 235	. . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆)
28		prct 32805	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
29	5, 7, 28	syl2anc 590	. . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
30		disjdifprg 32664	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦)
31	5, 4, 30	syl2anc 590	. . . . 5 ⊢ (𝜑 → Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦)
32		prcom 4664	. . . . . . 7 ⊢ {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)}
33	32	a1i 11	. . . . . 6 ⊢ (𝜑 → {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)})
34	33	disjeq1d 5047	. . . . 5 ⊢ (𝜑 → (Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦 ↔ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦))
35	31, 34	mpbid 233	. . . 4 ⊢ (𝜑 → Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)
36		measvun 34393	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ {𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆 ∧ ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)) → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦))
37	1, 27, 29, 35, 36	syl112anc 1382	. . 3 ⊢ (𝜑 → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦))
38		uniprg 4854	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴)))
39	5, 7, 38	syl2anc 590	. . . . 5 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴)))
40		measssd.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
41		undif 4410	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
42	40, 41	sylib 219	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
43	39, 42	eqtrd 2774	. . . 4 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
44	43	fveq2d 6831	. . 3 ⊢ (𝜑 → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = (𝑀‘𝐵))
45		fveq2 6827	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑀‘𝑦) = (𝑀‘𝐴))
46	45	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑀‘𝑦) = (𝑀‘𝐴))
47		fveq2 6827	. . . . 5 ⊢ (𝑦 = (𝐵 ∖ 𝐴) → (𝑀‘𝑦) = (𝑀‘(𝐵 ∖ 𝐴)))
48	47	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐵 ∖ 𝐴)) → (𝑀‘𝑦) = (𝑀‘(𝐵 ∖ 𝐴)))
49		eqimss 3973	. . . . . . . . . 10 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 ⊆ (𝐵 ∖ 𝐴))
50		ssdifeq0 4414	. . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
51	49, 50	sylib 219	. . . . . . . . 9 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 = ∅)
52	51	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → 𝐴 = ∅)
53	52	fveq2d 6831	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘𝐴) = (𝑀‘∅))
54		measvnul 34390	. . . . . . . . 9 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
55	1, 54	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘∅) = 0)
56	55	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘∅) = 0)
57	53, 56	eqtrd 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘𝐴) = 0)
58	57	orcd 879	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → ((𝑀‘𝐴) = 0 ∨ (𝑀‘𝐴) = +∞))
59	58	ex 413	. . . 4 ⊢ (𝜑 → (𝐴 = (𝐵 ∖ 𝐴) → ((𝑀‘𝐴) = 0 ∨ (𝑀‘𝐴) = +∞)))
60	46, 48, 5, 7, 14, 9, 59	esumpr2 34251	. . 3 ⊢ (𝜑 → Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦) = ((𝑀‘𝐴) +_𝑒 (𝑀‘(𝐵 ∖ 𝐴))))
61	37, 44, 60	3eqtr3d 2782	. 2 ⊢ (𝜑 → (𝑀‘𝐵) = ((𝑀‘𝐴) +_𝑒 (𝑀‘(𝐵 ∖ 𝐴))))
62	22, 61	breqtrrd 5100	1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 ∅c0 4261 𝒫 cpw 4529 {cpr 4557 ∪ cuni 4838 Disj wdisj 5039 class class class wbr 5072 ran crn 5619 ‘cfv 6485 (class class class)co 7356 ωcom 7806 ≼ cdom 8881 0cc0 11029 +∞cpnf 11167 ℝ^*cxr 11169 ≤ cle 11171 +_𝑒 cxad 13052 [,]cicc 13292 Σ^*cesum 34211 sigAlgebracsiga 34292 measurescmeas 34379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-ac2 10376 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-disj 5040 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9816 df-card 9854 df-acn 9857 df-ac 10029 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-ordt 17456 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-ps 18523 df-tsr 18524 df-plusf 18598 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-subrng 20518 df-subrg 20542 df-abv 20781 df-lmod 20852 df-scaf 20853 df-sra 21163 df-rgmod 21164 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-tmd 24055 df-tgp 24056 df-tsms 24110 df-trg 24143 df-xms 24303 df-ms 24304 df-tms 24305 df-nm 24565 df-ngp 24566 df-nrg 24568 df-nlm 24569 df-ii 24862 df-cncf 24863 df-limc 25851 df-dv 25852 df-log 26538 df-esum 34212 df-siga 34293 df-meas 34380

This theorem is referenced by: measiun 34402 aean 34428 sibfinima 34523 prob01 34597 probinc 34605 probmeasb 34614 cndprob01 34619 dstfrvinc 34661

W3C validator