measxun2 - Mathbox for Thierry Arnoux

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Theorem measxun2 34191

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 1135	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝑀 ∈ (measures‘𝑆))
2		simp2r 1199	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑆)
3		measbase 34178	. . . . . 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 1198	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐴 ∈ 𝑆)
6		difelsiga 34114	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)
7	4, 5, 2, 6	syl3anc 1370	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ 𝑆)
8		prelpwi 5458	. . . 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 32732	. . . . 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 1137	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
13		disjdifprg2 32596	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → Disj 𝑥 ∈ {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)}𝑥)
14		prcom 4737	. . . . . . . . 9 ⊢ {(𝐴 ∖ 𝐵), 𝐵} = {𝐵, (𝐴 ∖ 𝐵)}
15		dfss 3982	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐵 ∩ 𝐴))
16	15	biimpi 216	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))
17		incom 4217	. . . . . . . . . . 11 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
18	16, 17	eqtrdi 2791	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (𝐴 ∩ 𝐵))
19	18	preq2d 4745	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → {(𝐴 ∖ 𝐵), 𝐵} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)})
20	14, 19	eqtr3id 2789	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → {𝐵, (𝐴 ∖ 𝐵)} = {(𝐴 ∖ 𝐵), (𝐴 ∩ 𝐵)})
21	20	disjeq1d 5123	. . . . . . 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 34190	. . 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 1370	. 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 4928	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) → ∪ {𝐵, (𝐴 ∖ 𝐵)} = (𝐵 ∪ (𝐴 ∖ 𝐵)))
30		undif 4488	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
31	30	biimpi 216	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
32	29, 31	sylan9eq 2795	. . . 4 ⊢ (((𝐵 ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) → ∪ {𝐵, (𝐴 ∖ 𝐵)} = 𝐴)
33	32	fveq2d 6911	. . 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 6911	. . 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 6911	. . 3 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = (𝐴 ∖ 𝐵)) → (𝑀‘𝑥) = (𝑀‘(𝐴 ∖ 𝐵)))
39		measvxrge0 34186	. . . 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 34186	. . . 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 4054	. . . . . . . . 9 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → 𝐵 ⊆ (𝐴 ∖ 𝐵))
44		ssdifeq0 4493	. . . . . . . . 9 ⊢ (𝐵 ⊆ (𝐴 ∖ 𝐵) ↔ 𝐵 = ∅)
45	43, 44	sylib 218	. . . . . . . 8 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → 𝐵 = ∅)
46	45	fveq2d 6911	. . . . . . 7 ⊢ (𝐵 = (𝐴 ∖ 𝐵) → (𝑀‘𝐵) = (𝑀‘∅))
47		measvnul 34187	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
48	46, 47	sylan9eqr 2797	. . . . . 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 34048	. 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 2783	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 1537 ∈ wcel 2106 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {cpr 4633 ∪ cuni 4912 Disj wdisj 5115 class class class wbr 5148 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ωcom 7887 ≼ cdom 8982 0cc0 11153 +∞cpnf 11290 +_𝑒 cxad 13150 [,]cicc 13387 Σ^*cesum 34008 sigAlgebracsiga 34089 measurescmeas 34176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-ordt 17548 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-ps 18624 df-tsr 18625 df-plusf 18665 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-abv 20827 df-lmod 20877 df-scaf 20878 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-tmd 24096 df-tgp 24097 df-tsms 24151 df-trg 24184 df-xms 24346 df-ms 24347 df-tms 24348 df-nm 24611 df-ngp 24612 df-nrg 24614 df-nlm 24615 df-ii 24917 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-esum 34009 df-siga 34090 df-meas 34177

This theorem is referenced by: measun 34192

W3C validator