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Theorem measssd 34372

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 34354	. . . . . . 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 34290	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵 ∖ 𝐴) ∈ 𝑆)
7	3, 4, 5, 6	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑆)
8		measvxrge0 34362	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞))
9	1, 7, 8	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞))
10		elxrge0 13373	. . . . 5 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ ℝ^* ∧ 0 ≤ (𝑀‘(𝐵 ∖ 𝐴))))
11	10	simprbi 496	. . . 4 ⊢ ((𝑀‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞) → 0 ≤ (𝑀‘(𝐵 ∖ 𝐴)))
12	9, 11	syl 17	. . 3 ⊢ (𝜑 → 0 ≤ (𝑀‘(𝐵 ∖ 𝐴)))
13		measvxrge0 34362	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞))
14	1, 5, 13	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞))
15		elxrge0 13373	. . . . . 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 32837	. . . 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 584	. . 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 4777	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆)
24	5, 7, 23	syl2anc 584	. . . . 5 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆)
25		prex 5382	. . . . . 6 ⊢ {𝐴, (𝐵 ∖ 𝐴)} ∈ V
26	25	elpw 4558	. . . . 5 ⊢ ({𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆 ↔ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆)
27	24, 26	sylibr 234	. . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆)
28		prct 32792	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
29	5, 7, 28	syl2anc 584	. . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
30		disjdifprg 32650	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦)
31	5, 4, 30	syl2anc 584	. . . . 5 ⊢ (𝜑 → Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦)
32		prcom 4689	. . . . . . 7 ⊢ {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)}
33	32	a1i 11	. . . . . 6 ⊢ (𝜑 → {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)})
34	33	disjeq1d 5073	. . . . 5 ⊢ (𝜑 → (Disj 𝑦 ∈ {(𝐵 ∖ 𝐴), 𝐴}𝑦 ↔ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦))
35	31, 34	mpbid 232	. . . 4 ⊢ (𝜑 → Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)
36		measvun 34366	. . . 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 1376	. . 3 ⊢ (𝜑 → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦))
38		uniprg 4879	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴)))
39	5, 7, 38	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴)))
40		measssd.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
41		undif 4434	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
42	40, 41	sylib 218	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
43	39, 42	eqtrd 2771	. . . 4 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
44	43	fveq2d 6838	. . 3 ⊢ (𝜑 → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = (𝑀‘𝐵))
45		fveq2 6834	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑀‘𝑦) = (𝑀‘𝐴))
46	45	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑀‘𝑦) = (𝑀‘𝐴))
47		fveq2 6834	. . . . 5 ⊢ (𝑦 = (𝐵 ∖ 𝐴) → (𝑀‘𝑦) = (𝑀‘(𝐵 ∖ 𝐴)))
48	47	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝐵 ∖ 𝐴)) → (𝑀‘𝑦) = (𝑀‘(𝐵 ∖ 𝐴)))
49		eqimss 3992	. . . . . . . . . 10 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 ⊆ (𝐵 ∖ 𝐴))
50		ssdifeq0 4439	. . . . . . . . . 10 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
51	49, 50	sylib 218	. . . . . . . . 9 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 = ∅)
52	51	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → 𝐴 = ∅)
53	52	fveq2d 6838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘𝐴) = (𝑀‘∅))
54		measvnul 34363	. . . . . . . . 9 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
55	1, 54	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘∅) = 0)
56	55	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘∅) = 0)
57	53, 56	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘𝐴) = 0)
58	57	orcd 873	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → ((𝑀‘𝐴) = 0 ∨ (𝑀‘𝐴) = +∞))
59	58	ex 412	. . . 4 ⊢ (𝜑 → (𝐴 = (𝐵 ∖ 𝐴) → ((𝑀‘𝐴) = 0 ∨ (𝑀‘𝐴) = +∞)))
60	46, 48, 5, 7, 14, 9, 59	esumpr2 34224	. . 3 ⊢ (𝜑 → Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦) = ((𝑀‘𝐴) +_𝑒 (𝑀‘(𝐵 ∖ 𝐴))))
61	37, 44, 60	3eqtr3d 2779	. 2 ⊢ (𝜑 → (𝑀‘𝐵) = ((𝑀‘𝐴) +_𝑒 (𝑀‘(𝐵 ∖ 𝐴))))
62	22, 61	breqtrrd 5126	1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ∖ cdif 3898 ∪ cun 3899 ⊆ wss 3901 ∅c0 4285 𝒫 cpw 4554 {cpr 4582 ∪ cuni 4863 Disj wdisj 5065 class class class wbr 5098 ran crn 5625 ‘cfv 6492 (class class class)co 7358 ωcom 7808 ≼ cdom 8881 0cc0 11026 +∞cpnf 11163 ℝ^*cxr 11165 ≤ cle 11167 +_𝑒 cxad 13024 [,]cicc 13264 Σ^*cesum 34184 sigAlgebracsiga 34265 measurescmeas 34352

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-ac2 10373 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 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 9813 df-card 9851 df-acn 9854 df-ac 10026 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-cos 15993 df-pi 15995 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-ordt 17422 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-ps 18489 df-tsr 18490 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-subrng 20479 df-subrg 20503 df-abv 20742 df-lmod 20813 df-scaf 20814 df-sra 21125 df-rgmod 21126 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-tmd 24016 df-tgp 24017 df-tsms 24071 df-trg 24104 df-xms 24264 df-ms 24265 df-tms 24266 df-nm 24526 df-ngp 24527 df-nrg 24529 df-nlm 24530 df-ii 24826 df-cncf 24827 df-limc 25823 df-dv 25824 df-log 26521 df-esum 34185 df-siga 34266 df-meas 34353

This theorem is referenced by: measiun 34375 aean 34401 sibfinima 34496 prob01 34570 probinc 34578 probmeasb 34587 cndprob01 34592 dstfrvinc 34634

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