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Theorem measssd 34321

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 34303	. . . . . . 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 34239	. . . . . 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 34311	. . . . 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 13371	. . . . 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 34311	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞))
14	1, 5, 13	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞))
15		elxrge0 13371	. . . . . 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 32786	. . . 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 4775	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆)
24	5, 7, 23	syl2anc 584	. . . . 5 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆)
25		prex 5380	. . . . . 6 ⊢ {𝐴, (𝐵 ∖ 𝐴)} ∈ V
26	25	elpw 4556	. . . . 5 ⊢ ({𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆 ↔ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑆)
27	24, 26	sylibr 234	. . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ∈ 𝒫 𝑆)
28		prct 32741	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∖ 𝐴) ∈ 𝑆) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
29	5, 7, 28	syl2anc 584	. . . 4 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
30		disjdifprg 32599	. . . . . 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 4687	. . . . . . 7 ⊢ {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)}
33	32	a1i 11	. . . . . 6 ⊢ (𝜑 → {(𝐵 ∖ 𝐴), 𝐴} = {𝐴, (𝐵 ∖ 𝐴)})
34	33	disjeq1d 5071	. . . . 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 34315	. . . 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 4877	. . . . . 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 4432	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
42	40, 41	sylib 218	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
43	39, 42	eqtrd 2769	. . . 4 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
44	43	fveq2d 6836	. . 3 ⊢ (𝜑 → (𝑀‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = (𝑀‘𝐵))
45		fveq2 6832	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑀‘𝑦) = (𝑀‘𝐴))
46	45	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑀‘𝑦) = (𝑀‘𝐴))
47		fveq2 6832	. . . . 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 3990	. . . . . . . . . 10 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 ⊆ (𝐵 ∖ 𝐴))
50		ssdifeq0 4437	. . . . . . . . . 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 6836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑀‘𝐴) = (𝑀‘∅))
54		measvnul 34312	. . . . . . . . 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 2769	. . . . . 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 34173	. . 3 ⊢ (𝜑 → Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑀‘𝑦) = ((𝑀‘𝐴) +_𝑒 (𝑀‘(𝐵 ∖ 𝐴))))
61	37, 44, 60	3eqtr3d 2777	. 2 ⊢ (𝜑 → (𝑀‘𝐵) = ((𝑀‘𝐴) +_𝑒 (𝑀‘(𝐵 ∖ 𝐴))))
62	22, 61	breqtrrd 5124	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 3896 ∪ cun 3897 ⊆ wss 3899 ∅c0 4283 𝒫 cpw 4552 {cpr 4580 ∪ cuni 4861 Disj wdisj 5063 class class class wbr 5096 ran crn 5623 ‘cfv 6490 (class class class)co 7356 ωcom 7806 ≼ cdom 8879 0cc0 11024 +∞cpnf 11161 ℝ^*cxr 11163 ≤ cle 11165 +_𝑒 cxad 13022 [,]cicc 13262 Σ^*cesum 34133 sigAlgebracsiga 34214 measurescmeas 34301

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 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-ac2 10371 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103 ax-mulf 11104

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-disj 5064 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 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 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-dju 9811 df-card 9849 df-acn 9852 df-ac 10024 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-fac 14195 df-bc 14224 df-hash 14252 df-shft 14988 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-sum 15608 df-ef 15988 df-sin 15990 df-cos 15991 df-pi 15993 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-ordt 17420 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-ps 18487 df-tsr 18488 df-plusf 18562 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18996 df-subg 19051 df-cntz 19244 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-subrng 20477 df-subrg 20501 df-abv 20740 df-lmod 20811 df-scaf 20812 df-sra 21123 df-rgmod 21124 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-tmd 24014 df-tgp 24015 df-tsms 24069 df-trg 24102 df-xms 24262 df-ms 24263 df-tms 24264 df-nm 24524 df-ngp 24525 df-nrg 24527 df-nlm 24528 df-ii 24824 df-cncf 24825 df-limc 25821 df-dv 25822 df-log 26519 df-esum 34134 df-siga 34215 df-meas 34302

This theorem is referenced by: measiun 34324 aean 34350 sibfinima 34445 prob01 34519 probinc 34527 probmeasb 34536 cndprob01 34541 dstfrvinc 34583

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