measxun2 - Mathbox for Thierry Arnoux

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Theorem measxun2 34074

Description: The measure the union of two complementary sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.)

**Assertion**
Ref	Expression
measxun2	⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘𝐴) = ((𝑀‘𝐵) +_𝑒 (𝑀‘(𝐴 ∖ 𝐵))))

Proof of Theorem measxun2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1133	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝑀 ∈ (measures‘𝑆))
2		simp2r 1197	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑆)
3		measbase 34061	. . . . . 6 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
4	1, 3	syl 17	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝑆 ∈ ∪ ran sigAlgebra)
5		simp2l 1196	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐴 ∈ 𝑆)
6		difelsiga 33997	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)
7	4, 5, 2, 6	syl3anc 1368	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ 𝑆)
8		prelpwi 5455	. . . 4 ⊢ ((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → {𝐵, (𝐴 ∖ 𝐵)} ∈ 𝒫 𝑆)
9	2, 7, 8	syl2anc 582	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → {𝐵, (𝐴 ∖ 𝐵)} ∈ 𝒫 𝑆)
10		prct 32628	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → {𝐵, (𝐴 ∖ 𝐵)} ≼ ω)
11	2, 7, 10	syl2anc 582	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → {𝐵, (𝐴 ∖ 𝐵)} ≼ ω)
12		simp3 1135	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
13		disjdifprg2 32497	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥)
14		prcom 4742	. . . . . . . . 9 ⊢ {(𝐴 ∖ 𝐵), 𝐵} = {𝐵, (𝐴 ∖ 𝐵)}
15		dfss 3977	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐵 ∩ 𝐴))
16	15	biimpi 215	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))
17		incom 4210	. . . . . . . . . . 11 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
18	16, 17	eqtrdi 2785	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (𝐴 ∩ 𝐵))
19	18	preq2d 4750	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → {(𝐴 ∖ 𝐵), 𝐵} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)})
20	14, 19	eqtr3id 2783	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → {𝐵, (𝐴 ∖ 𝐵)} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)})
21	20	disjeq1d 5129	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥 ↔ Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥))
22	21	biimprd 247	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥 → Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥))
23	13, 22	mpan9 505	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ⊆ 𝐴) → Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥)
24	5, 12, 23	syl2anc 582	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥)
25	11, 24	jca 510	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → ({𝐵, (𝐴 ∖ 𝐵)} ≼ ω ∧ Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥))
26		measvun 34073	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ {𝐵, (𝐴 ∖ 𝐵)} ∈ 𝒫 𝑆 ∧ ({𝐵, (𝐴 ∖ 𝐵)} ≼ ω ∧ Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥)) → (𝑀‘∪ {𝐵, (𝐴 ∖ 𝐵)}) = Σ^*𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)} (𝑀‘𝑥))
27	1, 9, 25, 26	syl3anc 1368	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘∪ {𝐵, (𝐴 ∖ 𝐵)}) = Σ^*𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)} (𝑀‘𝑥))
28	2, 7	jca 510	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆))
29		uniprg 4932	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → ∪ {𝐵, (𝐴 ∖ 𝐵)} = (𝐵 ∪ (𝐴 ∖ 𝐵)))
30		undif 4487	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
31	30	biimpi 215	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
32	29, 31	sylan9eq 2789	. . . 4 ⊢ (((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → ∪ {𝐵, (𝐴 ∖ 𝐵)} = 𝐴)
33	32	fveq2d 6907	. . 3 ⊢ (((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘∪ {𝐵, (𝐴 ∖ 𝐵)}) = (𝑀‘𝐴))
34	28, 12, 33	syl2anc 582	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘∪ {𝐵, (𝐴 ∖ 𝐵)}) = (𝑀‘𝐴))
35		simpr 483	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
36	35	fveq2d 6907	. . 3 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = 𝐵) → (𝑀‘𝑥) = (𝑀‘𝐵))
37		simpr 483	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = (𝐴 ∖ 𝐵)) → 𝑥 = (𝐴 ∖ 𝐵))
38	37	fveq2d 6907	. . 3 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = (𝐴 ∖ 𝐵)) → (𝑀‘𝑥) = (𝑀‘(𝐴 ∖ 𝐵)))
39		measvxrge0 34069	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ (0[,]+∞))
40	1, 2, 39	syl2anc 582	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘𝐵) ∈ (0[,]+∞))
41		measvxrge0 34069	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → (𝑀‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞))
42	1, 7, 41	syl2anc 582	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞))
43		eqimss 4049	. . . . . . . . 9 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → 𝐵 ⊆ (𝐴 ∖ 𝐵))
44		ssdifeq0 4492	. . . . . . . . 9 ⊢ (𝐵 ⊆ (𝐴 ∖ 𝐵) ↔ 𝐵 = ∅)
45	43, 44	sylib 217	. . . . . . . 8 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → 𝐵 = ∅)
46	45	fveq2d 6907	. . . . . . 7 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → (𝑀‘𝐵) = (𝑀‘∅))
47		measvnul 34070	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
48	46, 47	sylan9eqr 2791	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 = (𝐴 ∖ 𝐵)) → (𝑀‘𝐵) = 0)
49	1, 48	sylan 578	. . . . 5 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝐵 = (𝐴 ∖ 𝐵)) → (𝑀‘𝐵) = 0)
50	49	orcd 871	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝐵 = (𝐴 ∖ 𝐵)) → ((𝑀‘𝐵) = 0 ∨ (𝑀‘𝐵) = +∞))
51	50	ex 411	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝐵 = (𝐴 ∖ 𝐵) → ((𝑀‘𝐵) = 0 ∨ (𝑀‘𝐵) = +∞)))
52	36, 38, 2, 7, 40, 42, 51	esumpr2 33931	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → Σ^*𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)} (𝑀‘𝑥) = ((𝑀‘𝐵) +_𝑒 (𝑀‘(𝐴 ∖ 𝐵))))
53	27, 34, 52	3eqtr3d 2777	1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘𝐴) = ((𝑀‘𝐵) +_𝑒 (𝑀‘(𝐴 ∖ 𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∈ wcel 2100 ∖ cdif 3955 ∪ cun 3956 ∩ cin 3957 ⊆ wss 3958 ∅c0 4333 𝒫 cpw 4608 {cpr 4636 ∪ cuni 4916 Disj wdisj 5121 class class class wbr 5154 ran crn 5685 ‘cfv 6556 (class class class)co 7427 ωcom 7881 ≼ cdom 8976 0cc0 11159 +∞cpnf 11296 +_𝑒 cxad 13148 [,]cicc 13385 Σ^*cesum 33891 sigAlgebracsiga 33972 measurescmeas 34059

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2102 ax-9 2110 ax-10 2133 ax-11 2150 ax-12 2170 ax-ext 2700 ax-rep 5291 ax-sep 5305 ax-nul 5312 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-inf2 9685 ax-ac2 10507 ax-cnex 11215 ax-resscn 11216 ax-1cn 11217 ax-icn 11218 ax-addcl 11219 ax-addrcl 11220 ax-mulcl 11221 ax-mulrcl 11222 ax-mulcom 11223 ax-addass 11224 ax-mulass 11225 ax-distr 11226 ax-i2m1 11227 ax-1ne0 11228 ax-1rid 11229 ax-rnegex 11230 ax-rrecex 11231 ax-cnre 11232 ax-pre-lttri 11233 ax-pre-lttrn 11234 ax-pre-ltadd 11235 ax-pre-mulgt0 11236 ax-pre-sup 11237 ax-addf 11238 ax-mulf 11239

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2062 df-mo 2532 df-eu 2561 df-clab 2707 df-cleq 2721 df-clel 2806 df-nfc 2881 df-ne 2934 df-nel 3040 df-ral 3055 df-rex 3064 df-rmo 3373 df-reu 3374 df-rab 3429 df-v 3474 df-sbc 3788 df-csb 3904 df-dif 3961 df-un 3963 df-in 3965 df-ss 3975 df-pss 3978 df-nul 4334 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4917 df-int 4958 df-iun 5006 df-iin 5007 df-disj 5122 df-br 5155 df-opab 5217 df-mpt 5238 df-tr 5272 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-se 5640 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6315 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7383 df-ov 7430 df-oprab 7431 df-mpo 7432 df-of 7692 df-om 7882 df-1st 8008 df-2nd 8009 df-supp 8180 df-frecs 8300 df-wrecs 8331 df-recs 8405 df-rdg 8444 df-1o 8500 df-2o 8501 df-er 8738 df-map 8861 df-pm 8862 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-fsupp 9407 df-fi 9455 df-sup 9486 df-inf 9487 df-oi 9554 df-dju 9945 df-card 9983 df-acn 9986 df-ac 10160 df-pnf 11301 df-mnf 11302 df-xr 11303 df-ltxr 11304 df-le 11305 df-sub 11497 df-neg 11498 df-div 11923 df-nn 12269 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12529 df-z 12615 df-dec 12734 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13150 df-xadd 13151 df-xmul 13152 df-ioo 13386 df-ioc 13387 df-ico 13388 df-icc 13389 df-fz 13543 df-fzo 13686 df-fl 13817 df-mod 13895 df-seq 14027 df-exp 14087 df-fac 14297 df-bc 14326 df-hash 14354 df-shft 15085 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-limsup 15486 df-clim 15503 df-rlim 15504 df-sum 15704 df-ef 16082 df-sin 16084 df-cos 16085 df-pi 16087 df-struct 17162 df-sets 17179 df-slot 17197 df-ndx 17209 df-base 17227 df-ress 17256 df-plusg 17292 df-mulr 17293 df-starv 17294 df-sca 17295 df-vsca 17296 df-ip 17297 df-tset 17298 df-ple 17299 df-ds 17301 df-unif 17302 df-hom 17303 df-cco 17304 df-rest 17450 df-topn 17451 df-0g 17469 df-gsum 17470 df-topgen 17471 df-pt 17472 df-prds 17475 df-ordt 17529 df-xrs 17530 df-qtop 17535 df-imas 17536 df-xps 17538 df-mre 17612 df-mrc 17613 df-acs 17615 df-ps 18604 df-tsr 18605 df-plusf 18645 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-mhm 18786 df-submnd 18787 df-grp 18944 df-minusg 18945 df-sbg 18946 df-mulg 19076 df-subg 19131 df-cntz 19325 df-cmn 19792 df-abl 19793 df-mgp 20130 df-rng 20148 df-ur 20177 df-ring 20230 df-cring 20231 df-subrng 20540 df-subrg 20565 df-abv 20800 df-lmod 20850 df-scaf 20851 df-sra 21163 df-rgmod 21164 df-psmet 21347 df-xmet 21348 df-met 21349 df-bl 21350 df-mopn 21351 df-fbas 21352 df-fg 21353 df-cnfld 21356 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cn 23222 df-cnp 23223 df-haus 23310 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-tmd 24067 df-tgp 24068 df-tsms 24122 df-trg 24155 df-xms 24317 df-ms 24318 df-tms 24319 df-nm 24582 df-ngp 24583 df-nrg 24585 df-nlm 24586 df-ii 24888 df-cncf 24889 df-limc 25886 df-dv 25887 df-log 26580 df-esum 33892 df-siga 33973 df-meas 34060

This theorem is referenced by: measun 34075

W3C validator