measxun2 - Mathbox for Thierry Arnoux

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Theorem measxun2 29600

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 1054	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝑀 ∈ (measures‘𝑆))
2		simp2r 1081	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑆)
3		measbase 29587	. . . . . 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 1080	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐴 ∈ 𝑆)
6		difelsiga 29523	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)
7	4, 5, 2, 6	syl3anc 1318	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ 𝑆)
8		prelpwi 4842	. . . 4 ⊢ ((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → {𝐵, (𝐴 ∖ 𝐵)} ∈ 𝒫 𝑆)
9	2, 7, 8	syl2anc 691	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → {𝐵, (𝐴 ∖ 𝐵)} ∈ 𝒫 𝑆)
10		prct 28875	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → {𝐵, (𝐴 ∖ 𝐵)} ≼ ω)
11	2, 7, 10	syl2anc 691	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → {𝐵, (𝐴 ∖ 𝐵)} ≼ ω)
12		simp3 1056	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
13		disjdifprg2 28771	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥)
14		prcom 4211	. . . . . . . . 9 ⊢ {(𝐴 ∖ 𝐵), 𝐵} = {𝐵, (𝐴 ∖ 𝐵)}
15		dfss 3555	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐵 ∩ 𝐴))
16	15	biimpi 205	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))
17		incom 3767	. . . . . . . . . . 11 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
18	16, 17	syl6eq 2660	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (𝐴 ∩ 𝐵))
19	18	preq2d 4219	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → {(𝐴 ∖ 𝐵), 𝐵} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)})
20	14, 19	syl5eqr 2658	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → {𝐵, (𝐴 ∖ 𝐵)} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)})
21	20	disjeq1d 4561	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥 ↔ Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥))
22	21	biimprd 237	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → (Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥 → Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥))
23	13, 22	mpan9 485	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ⊆ 𝐴) → Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥)
24	5, 12, 23	syl2anc 691	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥)
25	11, 24	jca 553	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → ({𝐵, (𝐴 ∖ 𝐵)} ≼ ω ∧ Disj 𝑥 ∈ {𝐵, (𝐴 ∖ 𝐵)}𝑥))
26		measvun 29599	. . 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 1318	. 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 553	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆))
29		uniprg 4386	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → ∪ {𝐵, (𝐴 ∖ 𝐵)} = (𝐵 ∪ (𝐴 ∖ 𝐵)))
30		undif 4001	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
31	30	biimpi 205	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
32	29, 31	sylan9eq 2664	. . . 4 ⊢ (((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → ∪ {𝐵, (𝐴 ∖ 𝐵)} = 𝐴)
33	32	fveq2d 6107	. . 3 ⊢ (((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘∪ {𝐵, (𝐴 ∖ 𝐵)}) = (𝑀‘𝐴))
34	28, 12, 33	syl2anc 691	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘∪ {𝐵, (𝐴 ∖ 𝐵)}) = (𝑀‘𝐴))
35		simpr 476	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
36	35	fveq2d 6107	. . 3 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = 𝐵) → (𝑀‘𝑥) = (𝑀‘𝐵))
37		simpr 476	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = (𝐴 ∖ 𝐵)) → 𝑥 = (𝐴 ∖ 𝐵))
38	37	fveq2d 6107	. . 3 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = (𝐴 ∖ 𝐵)) → (𝑀‘𝑥) = (𝑀‘(𝐴 ∖ 𝐵)))
39		measvxrge0 29595	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ (0[,]+∞))
40	1, 2, 39	syl2anc 691	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘𝐵) ∈ (0[,]+∞))
41		measvxrge0 29595	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → (𝑀‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞))
42	1, 7, 41	syl2anc 691	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝑀‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞))
43		eqimss 3620	. . . . . . . . 9 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → 𝐵 ⊆ (𝐴 ∖ 𝐵))
44		ssdifeq0 4003	. . . . . . . . 9 ⊢ (𝐵 ⊆ (𝐴 ∖ 𝐵) ↔ 𝐵 = ∅)
45	43, 44	sylib 207	. . . . . . . 8 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → 𝐵 = ∅)
46	45	fveq2d 6107	. . . . . . 7 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → (𝑀‘𝐵) = (𝑀‘∅))
47		measvnul 29596	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
48	46, 47	sylan9eqr 2666	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 = (𝐴 ∖ 𝐵)) → (𝑀‘𝐵) = 0)
49	1, 48	sylan 487	. . . . 5 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝐵 = (𝐴 ∖ 𝐵)) → (𝑀‘𝐵) = 0)
50	49	orcd 406	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝐵 = (𝐴 ∖ 𝐵)) → ((𝑀‘𝐵) = 0 ∨ (𝑀‘𝐵) = +∞))
51	50	ex 449	. . 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 29456	. 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 2652	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 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {cpr 4127 ∪ cuni 4372 Disj wdisj 4553 class class class wbr 4583 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ≼ cdom 7839 0cc0 9815 +∞cpnf 9950 +_𝑒 cxad 11820 [,]cicc 12049 Σ^*cesum 29416 sigAlgebracsiga 29497 measurescmeas 29585

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-ac2 9168 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-ac 8822 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-ordt 15984 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-ps 17023 df-tsr 17024 df-plusf 17064 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-abv 18640 df-lmod 18688 df-scaf 18689 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tmd 21686 df-tgp 21687 df-tsms 21740 df-trg 21773 df-xms 21935 df-ms 21936 df-tms 21937 df-nm 22197 df-ngp 22198 df-nrg 22200 df-nlm 22201 df-ii 22488 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-esum 29417 df-siga 29498 df-meas 29586

This theorem is referenced by: measun 29601

W3C validator