Theorem volun 25587

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 1204	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐴 ∈ dom vol)
2		mblss 25573	. . . . . . . 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 1205	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐵 ∈ dom vol)
5		mblss 25573	. . . . . . . 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 11153	. . . . . . . 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 780	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝐴) ∈ ℝ)
11		simprr 782	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝐵) ∈ ℝ)
12		ovolun 25541	. . . . . . . 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 849	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
14		ovollecl 25525	. . . . . . 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 1389	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
16		mblsplit 25574	. . . . . 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 1389	. . . . 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 1206	. . . . . 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 4178	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21		incom 4161	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
22	20, 21	uneq12i 4119	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴)) = (𝐴 ∪ (𝐴 ∩ 𝐵))
23		unabs 4217	. . . . . . . . . . 11 ⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴
24	22, 23	eqtri 2784	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴)) = 𝐴
25	19, 24	eqtri 2784	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐴) = 𝐴
26	25	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∩ 𝐴) = 𝐴)
27	26	fveq2d 6867	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (vol*‘((𝐴 ∪ 𝐵) ∩ 𝐴)) = (vol*‘𝐴))
28		uncom 4111	. . . . . . . . . . 11 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
29	28	difeq1i 4076	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴)
30		difun2 4434	. . . . . . . . . 10 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
31	29, 30	eqtri 2784	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐴) = (𝐵 ∖ 𝐴)
32	21	eqeq1i 2766	. . . . . . . . . 10 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)
33		disj3 4407	. . . . . . . . . 10 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ 𝐵 = (𝐵 ∖ 𝐴))
34	32, 33	sylbb1 239	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) = ∅ → 𝐵 = (𝐵 ∖ 𝐴))
35	31, 34	eqtr4id 2815	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐴) = 𝐵)
36	35	fveq2d 6867	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (vol*‘((𝐴 ∪ 𝐵) ∖ 𝐴)) = (vol*‘𝐵))
37	27, 36	oveq12d 7410	. . . . . 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 2796	. . . 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 25572	. . . . . 6 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
42	41	eleq1d 2846	. . . . 5 ⊢ (𝐴 ∈ dom vol → ((vol‘𝐴) ∈ ℝ ↔ (vol*‘𝐴) ∈ ℝ))
43		mblvol 25572	. . . . . 6 ⊢ (𝐵 ∈ dom vol → (vol‘𝐵) = (vol*‘𝐵))
44	43	eleq1d 2846	. . . . 5 ⊢ (𝐵 ∈ dom vol → ((vol‘𝐵) ∈ ℝ ↔ (vol*‘𝐵) ∈ ℝ))
45	42, 44	bi2anan9 647	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) ↔ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
46	45	3adant3 1144	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) ↔ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
47		unmbl 25579	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ∈ dom vol)
48		mblvol 25572	. . . . . 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 7411	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘𝐴) + (vol‘𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵)))
51	49, 50	eqeq12d 2777	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘(𝐴 ∪ 𝐵)) = ((vol‘𝐴) + (vol‘𝐵)) ↔ (vol*‘(𝐴 ∪ 𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
52	51	3adant3 1144	. . 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 296	. 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 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285   class class class wbr 5099  dom cdm 5645  ‘cfv 6517  (class class class)co 7392  ℝcr 11069   + caddc 11073   ≤ cle 11214  vol*covol 25504  volcvol 25505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-sup 9385  df-inf 9386  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-n0 12479  df-z 12566  df-uz 12837  df-q 12947  df-rp 12991  df-ioo 13350  df-ico 13352  df-icc 13353  df-fz 13510  df-fl 13799  df-seq 14012  df-exp 14072  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-abs 15246  df-ovol 25506  df-vol 25507

This theorem is referenced by:  volinun  25588  volfiniun  25589  volsup  25598  ovolioo  25610  ismblfin  38124  volioc  46510  volico  46521