Theorem volun 25502

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 1192	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐴 ∈ dom vol)
2		mblss 25488	. . . . . . . 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 1193	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐵 ∈ dom vol)
5		mblss 25488	. . . . . . . 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 4144	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴 ∪ 𝐵) ⊆ ℝ)
8		readdcl 11109	. . . . . . . 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 25456	. . . . . . . 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 25440	. . . . . . 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 1373	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
16		mblsplit 25489	. . . . . 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 1373	. . . . 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 1194	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴 ∩ 𝐵) = ∅)
19		indir 4238	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐴) = ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴))
20		inidm 4179	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21		incom 4161	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
22	20, 21	uneq12i 4118	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴)) = (𝐴 ∪ (𝐴 ∩ 𝐵))
23		unabs 4217	. . . . . . . . . . 11 ⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴
24	22, 23	eqtri 2759	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴)) = 𝐴
25	19, 24	eqtri 2759	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐴) = 𝐴
26	25	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∩ 𝐴) = 𝐴)
27	26	fveq2d 6838	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (vol*‘((𝐴 ∪ 𝐵) ∩ 𝐴)) = (vol*‘𝐴))
28		uncom 4110	. . . . . . . . . . 11 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
29	28	difeq1i 4074	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴)
30		difun2 4433	. . . . . . . . . 10 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
31	29, 30	eqtri 2759	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐴) = (𝐵 ∖ 𝐴)
32	21	eqeq1i 2741	. . . . . . . . . 10 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)
33		disj3 4406	. . . . . . . . . 10 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ 𝐵 = (𝐵 ∖ 𝐴))
34	32, 33	sylbb1 237	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) = ∅ → 𝐵 = (𝐵 ∖ 𝐴))
35	31, 34	eqtr4id 2790	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐴) = 𝐵)
36	35	fveq2d 6838	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (vol*‘((𝐴 ∪ 𝐵) ∖ 𝐴)) = (vol*‘𝐵))
37	27, 36	oveq12d 7376	. . . . . 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 2771	. . . 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 25487	. . . . . 6 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
42	41	eleq1d 2821	. . . . 5 ⊢ (𝐴 ∈ dom vol → ((vol‘𝐴) ∈ ℝ ↔ (vol*‘𝐴) ∈ ℝ))
43		mblvol 25487	. . . . . 6 ⊢ (𝐵 ∈ dom vol → (vol‘𝐵) = (vol*‘𝐵))
44	43	eleq1d 2821	. . . . 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 25494	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ∈ dom vol)
48		mblvol 25487	. . . . . 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 7377	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘𝐴) + (vol‘𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵)))
51	49, 50	eqeq12d 2752	. . . 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 1541   ∈ wcel 2113   ∖ cdif 3898   ∪ cun 3899   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285   class class class wbr 5098  dom cdm 5624  ‘cfv 6492  (class class class)co 7358  ℝcr 11025   + caddc 11029   ≤ cle 11167  vol*covol 25419  volcvol 25420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9345  df-inf 9346  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-q 12862  df-rp 12906  df-ioo 13265  df-ico 13267  df-icc 13268  df-fz 13424  df-fl 13712  df-seq 13925  df-exp 13985  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-ovol 25421  df-vol 25422

This theorem is referenced by:  volinun  25503  volfiniun  25504  volsup  25513  ovolioo  25525  ismblfin  37858  volioc  46212  volico  46223