Theorem volun 24149

Description: The Lebesgue measure function is finitely additive. (Contributed by Mario Carneiro, 18-Mar-2014.)

Assertion

Ref	Expression
volun	⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ)) → (vol‘(𝐴 ∪ 𝐵)) = ((vol‘𝐴) + (vol‘𝐵)))

Proof of Theorem volun

Step	Hyp	Ref	Expression
1		simpl1 1188	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐴 ∈ dom vol)
2		mblss 24135	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
3	1, 2	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐴 ⊆ ℝ)
4		simpl2 1189	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐵 ∈ dom vol)
5		mblss 24135	. . . . . . . 8 ⊢ (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
6	4, 5	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐵 ⊆ ℝ)
7	3, 6	unssd 4113	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴 ∪ 𝐵) ⊆ ℝ)
8		readdcl 10609	. . . . . . . 8 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
9	8	adantl 485	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
10		simprl 770	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝐴) ∈ ℝ)
11		simprr 772	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝐵) ∈ ℝ)
12		ovolun 24103	. . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
13	3, 10, 6, 11, 12	syl22anc 837	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
14		ovollecl 24087	. . . . . . 7 ⊢ (((𝐴 ∪ 𝐵) ⊆ ℝ ∧ ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵))) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
15	7, 9, 13, 14	syl3anc 1368	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
16		mblsplit 24136	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ (𝐴 ∪ 𝐵) ⊆ ℝ ∧ (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ) → (vol*‘(𝐴 ∪ 𝐵)) = ((vol*‘((𝐴 ∪ 𝐵) ∩ 𝐴)) + (vol*‘((𝐴 ∪ 𝐵) ∖ 𝐴))))
17	1, 7, 15, 16	syl3anc 1368	. . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) = ((vol*‘((𝐴 ∪ 𝐵) ∩ 𝐴)) + (vol*‘((𝐴 ∪ 𝐵) ∖ 𝐴))))
18		simpl3 1190	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴 ∩ 𝐵) = ∅)
19		indir 4202	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐴) = ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴))
20		inidm 4145	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21		incom 4128	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
22	20, 21	uneq12i 4088	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴)) = (𝐴 ∪ (𝐴 ∩ 𝐵))
23		unabs 4181	. . . . . . . . . . 11 ⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴
24	22, 23	eqtri 2821	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴)) = 𝐴
25	19, 24	eqtri 2821	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐴) = 𝐴
26	25	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∩ 𝐴) = 𝐴)
27	26	fveq2d 6649	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (vol*‘((𝐴 ∪ 𝐵) ∩ 𝐴)) = (vol*‘𝐴))
28		uncom 4080	. . . . . . . . . . 11 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
29	28	difeq1i 4046	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴)
30		difun2 4387	. . . . . . . . . 10 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
31	29, 30	eqtri 2821	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐴) = (𝐵 ∖ 𝐴)
32	21	eqeq1i 2803	. . . . . . . . . 10 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)
33		disj3 4361	. . . . . . . . . 10 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ 𝐵 = (𝐵 ∖ 𝐴))
34	32, 33	sylbb1 240	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) = ∅ → 𝐵 = (𝐵 ∖ 𝐴))
35	31, 34	eqtr4id 2852	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐴) = 𝐵)
36	35	fveq2d 6649	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (vol*‘((𝐴 ∪ 𝐵) ∖ 𝐴)) = (vol*‘𝐵))
37	27, 36	oveq12d 7153	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((vol*‘((𝐴 ∪ 𝐵) ∩ 𝐴)) + (vol*‘((𝐴 ∪ 𝐵) ∖ 𝐴))) = ((vol*‘𝐴) + (vol*‘𝐵)))
38	18, 37	syl 17	. . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘((𝐴 ∪ 𝐵) ∩ 𝐴)) + (vol*‘((𝐴 ∪ 𝐵) ∖ 𝐴))) = ((vol*‘𝐴) + (vol*‘𝐵)))
39	17, 38	eqtrd 2833	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵)))
40	39	ex 416	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) → (((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘(𝐴 ∪ 𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
41		mblvol 24134	. . . . . 6 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
42	41	eleq1d 2874	. . . . 5 ⊢ (𝐴 ∈ dom vol → ((vol‘𝐴) ∈ ℝ ↔ (vol*‘𝐴) ∈ ℝ))
43		mblvol 24134	. . . . . 6 ⊢ (𝐵 ∈ dom vol → (vol‘𝐵) = (vol*‘𝐵))
44	43	eleq1d 2874	. . . . 5 ⊢ (𝐵 ∈ dom vol → ((vol‘𝐵) ∈ ℝ ↔ (vol*‘𝐵) ∈ ℝ))
45	42, 44	bi2anan9 638	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) ↔ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
46	45	3adant3 1129	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) ↔ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
47		unmbl 24141	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ∈ dom vol)
48		mblvol 24134	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ dom vol → (vol‘(𝐴 ∪ 𝐵)) = (vol*‘(𝐴 ∪ 𝐵)))
49	47, 48	syl 17	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (vol‘(𝐴 ∪ 𝐵)) = (vol*‘(𝐴 ∪ 𝐵)))
50	41, 43	oveqan12d 7154	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘𝐴) + (vol‘𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵)))
51	49, 50	eqeq12d 2814	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘(𝐴 ∪ 𝐵)) = ((vol‘𝐴) + (vol‘𝐵)) ↔ (vol*‘(𝐴 ∪ 𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
52	51	3adant3 1129	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) → ((vol‘(𝐴 ∪ 𝐵)) = ((vol‘𝐴) + (vol‘𝐵)) ↔ (vol*‘(𝐴 ∪ 𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
53	40, 46, 52	3imtr4d 297	. 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) → (vol‘(𝐴 ∪ 𝐵)) = ((vol‘𝐴) + (vol‘𝐵))))
54	53	imp 410	1 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ)) → (vol‘(𝐴 ∪ 𝐵)) = ((vol‘𝐴) + (vol‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243   class class class wbr 5030  dom cdm 5519  ‘cfv 6324  (class class class)co 7135  ℝcr 10525   + caddc 10529   ≤ cle 10665  vol*covol 24066  volcvol 24067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12886  df-fl 13157  df-seq 13365  df-exp 13426  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-ovol 24068  df-vol 24069

This theorem is referenced by:  volinun  24150  volfiniun  24151  volsup  24160  ovolioo  24172  ismblfin  35098  volioc  42614  volico  42625