measxun2 - Mathbox for Thierry Arnoux

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Theorem measxun2 34246

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 1136	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝑀 ∈ (measures‘𝑆))
2		simp2r 1201	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑆)
3		measbase 34233	. . . . . 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 1200	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐴 ∈ 𝑆)
6		difelsiga 34169	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)
7	4, 5, 2, 6	syl3anc 1373	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ 𝑆)
8		prelpwi 5427	. . . 4 ⊢ ((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → {𝐵, (𝐴 ∖ 𝐵)} ∈ 𝒫 𝑆)
9	2, 7, 8	syl2anc 584	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → {𝐵, (𝐴 ∖ 𝐵)} ∈ 𝒫 𝑆)
10		prct 32697	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → {𝐵, (𝐴 ∖ 𝐵)} ≼ ω)
11	2, 7, 10	syl2anc 584	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → {𝐵, (𝐴 ∖ 𝐵)} ≼ ω)
12		simp3 1138	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
13		disjdifprg2 32562	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥)
14		prcom 4713	. . . . . . . . 9 ⊢ {(𝐴 ∖ 𝐵), 𝐵} = {𝐵, (𝐴 ∖ 𝐵)}
15		dfss 3950	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐵 ∩ 𝐴))
16	15	biimpi 216	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))
17		incom 4189	. . . . . . . . . . 11 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
18	16, 17	eqtrdi 2787	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (𝐴 ∩ 𝐵))
19	18	preq2d 4721	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → {(𝐴 ∖ 𝐵), 𝐵} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)})
20	14, 19	eqtr3id 2785	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → {𝐵, (𝐴 ∖ 𝐵)} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)})
21	20	disjeq1d 5099	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥 ↔ Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥))
22	21	biimprd 248	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥 → Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥))
23	13, 22	mpan9 506	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ⊆ 𝐴) → Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥)
24	5, 12, 23	syl2anc 584	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥)
25	11, 24	jca 511	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → ({𝐵, (𝐴 ∖ 𝐵)} ≼ ω ∧ Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥))
26		measvun 34245	. . 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 1373	. 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 511	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆))
29		uniprg 4904	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → ∪ {𝐵, (𝐴 ∖ 𝐵)} = (𝐵 ∪ (𝐴 ∖ 𝐵)))
30		undif 4462	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
31	30	biimpi 216	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
32	29, 31	sylan9eq 2791	. . . 4 ⊢ (((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → ∪ {𝐵, (𝐴 ∖ 𝐵)} = 𝐴)
33	32	fveq2d 6885	. . 3 ⊢ (((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘∪ {𝐵, (𝐴 ∖ 𝐵)}) = (𝑀‘𝐴))
34	28, 12, 33	syl2anc 584	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘∪ {𝐵, (𝐴 ∖ 𝐵)}) = (𝑀‘𝐴))
35		simpr 484	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
36	35	fveq2d 6885	. . 3 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = 𝐵) → (𝑀‘𝑥) = (𝑀‘𝐵))
37		simpr 484	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = (𝐴 ∖ 𝐵)) → 𝑥 = (𝐴 ∖ 𝐵))
38	37	fveq2d 6885	. . 3 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = (𝐴 ∖ 𝐵)) → (𝑀‘𝑥) = (𝑀‘(𝐴 ∖ 𝐵)))
39		measvxrge0 34241	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ (0[,]+∞))
40	1, 2, 39	syl2anc 584	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘𝐵) ∈ (0[,]+∞))
41		measvxrge0 34241	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → (𝑀‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞))
42	1, 7, 41	syl2anc 584	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞))
43		eqimss 4022	. . . . . . . . 9 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → 𝐵 ⊆ (𝐴 ∖ 𝐵))
44		ssdifeq0 4467	. . . . . . . . 9 ⊢ (𝐵 ⊆ (𝐴 ∖ 𝐵) ↔ 𝐵 = ∅)
45	43, 44	sylib 218	. . . . . . . 8 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → 𝐵 = ∅)
46	45	fveq2d 6885	. . . . . . 7 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → (𝑀‘𝐵) = (𝑀‘∅))
47		measvnul 34242	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
48	46, 47	sylan9eqr 2793	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 = (𝐴 ∖ 𝐵)) → (𝑀‘𝐵) = 0)
49	1, 48	sylan 580	. . . . 5 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝐵 = (𝐴 ∖ 𝐵)) → (𝑀‘𝐵) = 0)
50	49	orcd 873	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝐵 = (𝐴 ∖ 𝐵)) → ((𝑀‘𝐵) = 0 ∨ (𝑀‘𝐵) = +∞))
51	50	ex 412	. . 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 34103	. 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 2779	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 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∖ cdif 3928 ∪ cun 3929 ∩ cin 3930 ⊆ wss 3931 ∅c0 4313 𝒫 cpw 4580 {cpr 4608 ∪ cuni 4888 Disj wdisj 5091 class class class wbr 5124 ran crn 5660 ‘cfv 6536 (class class class)co 7410 ωcom 7866 ≼ cdom 8962 0cc0 11134 +∞cpnf 11271 +_𝑒 cxad 13131 [,]cicc 13370 Σ^*cesum 34063 sigAlgebracsiga 34144 measurescmeas 34231

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-ac2 10482 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213 ax-mulf 11214

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-acn 9961 df-ac 10135 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-ef 16088 df-sin 16090 df-cos 16091 df-pi 16093 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-ordt 17520 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-ps 18581 df-tsr 18582 df-plusf 18622 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-cntz 19305 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-subrng 20511 df-subrg 20535 df-abv 20774 df-lmod 20824 df-scaf 20825 df-sra 21136 df-rgmod 21137 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-tmd 24015 df-tgp 24016 df-tsms 24070 df-trg 24103 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 25824 df-dv 25825 df-log 26522 df-esum 34064 df-siga 34145 df-meas 34232

This theorem is referenced by: measun 34247

W3C validator