Theorem volun 25516

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 1191	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐴 ∈ dom vol)
2		mblss 25502	. . . . . . . 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 1192	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐵 ∈ dom vol)
5		mblss 25502	. . . . . . . 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 4172	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴 ∪ 𝐵) ⊆ ℝ)
8		readdcl 11220	. . . . . . . 8 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
9	8	adantl 481	. . . . . . 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 25470	. . . . . . . 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 838	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
14		ovollecl 25454	. . . . . . 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 1372	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
16		mblsplit 25503	. . . . . 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 1372	. . . . 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 1193	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴 ∩ 𝐵) = ∅)
19		indir 4266	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐴) = ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴))
20		inidm 4207	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21		incom 4189	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
22	20, 21	uneq12i 4146	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴)) = (𝐴 ∪ (𝐴 ∩ 𝐵))
23		unabs 4245	. . . . . . . . . . 11 ⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴
24	22, 23	eqtri 2757	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴)) = 𝐴
25	19, 24	eqtri 2757	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐴) = 𝐴
26	25	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∩ 𝐴) = 𝐴)
27	26	fveq2d 6890	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (vol*‘((𝐴 ∪ 𝐵) ∩ 𝐴)) = (vol*‘𝐴))
28		uncom 4138	. . . . . . . . . . 11 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
29	28	difeq1i 4102	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴)
30		difun2 4461	. . . . . . . . . 10 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
31	29, 30	eqtri 2757	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐴) = (𝐵 ∖ 𝐴)
32	21	eqeq1i 2739	. . . . . . . . . 10 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)
33		disj3 4434	. . . . . . . . . 10 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ 𝐵 = (𝐵 ∖ 𝐴))
34	32, 33	sylbb1 237	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) = ∅ → 𝐵 = (𝐵 ∖ 𝐴))
35	31, 34	eqtr4id 2788	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐴) = 𝐵)
36	35	fveq2d 6890	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (vol*‘((𝐴 ∪ 𝐵) ∖ 𝐴)) = (vol*‘𝐵))
37	27, 36	oveq12d 7431	. . . . . 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 2769	. . . 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 412	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) → (((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘(𝐴 ∪ 𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
41		mblvol 25501	. . . . . 6 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
42	41	eleq1d 2818	. . . . 5 ⊢ (𝐴 ∈ dom vol → ((vol‘𝐴) ∈ ℝ ↔ (vol*‘𝐴) ∈ ℝ))
43		mblvol 25501	. . . . . 6 ⊢ (𝐵 ∈ dom vol → (vol‘𝐵) = (vol*‘𝐵))
44	43	eleq1d 2818	. . . . 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 1132	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) ↔ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
47		unmbl 25508	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ∈ dom vol)
48		mblvol 25501	. . . . . 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 7432	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘𝐴) + (vol‘𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵)))
51	49, 50	eqeq12d 2750	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘(𝐴 ∪ 𝐵)) = ((vol‘𝐴) + (vol‘𝐵)) ↔ (vol*‘(𝐴 ∪ 𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
52	51	3adant3 1132	. . 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 294	. 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 406	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ∖ cdif 3928   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313   class class class wbr 5123  dom cdm 5665  ‘cfv 6541  (class class class)co 7413  ℝcr 11136   + caddc 11140   ≤ cle 11278  vol*covol 25433  volcvol 25434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214  ax-pre-sup 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  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-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8727  df-map 8850  df-en 8968  df-dom 8969  df-sdom 8970  df-sup 9464  df-inf 9465  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-div 11903  df-nn 12249  df-2 12311  df-3 12312  df-n0 12510  df-z 12597  df-uz 12861  df-q 12973  df-rp 13017  df-ioo 13373  df-ico 13375  df-icc 13376  df-fz 13530  df-fl 13814  df-seq 14025  df-exp 14085  df-cj 15120  df-re 15121  df-im 15122  df-sqrt 15256  df-abs 15257  df-ovol 25435  df-vol 25436

This theorem is referenced by:  volinun  25517  volfiniun  25518  volsup  25527  ovolioo  25539  ismblfin  37627  volioc  45944  volico  45955