Theorem volun 25779

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 1210	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐴 ∈ dom vol)
2		mblss 25765	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
3	1, 2	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐴 ⊆ ℝ)
4		simpl2 1211	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐵 ∈ dom vol)
5		mblss 25765	. . . . . . . 8 ⊢ (𝐵 ∈ dom vol → 𝐵 ⊆ ℝ)
6	4, 5	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → 𝐵 ⊆ ℝ)
7	3, 6	unssd 4141	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴 ∪ 𝐵) ⊆ ℝ)
8		readdcl 11211	. . . . . . . 8 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
9	8	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ)
10		simprl 783	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝐴) ∈ ℝ)
11		simprr 785	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘𝐵) ∈ ℝ)
12		ovolun 25733	. . . . . . . 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 852	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)))
14		ovollecl 25717	. . . . . . 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 1398	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ)
16		mblsplit 25766	. . . . . 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 1398	. . . . 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 1212	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴 ∩ 𝐵) = ∅)
19		indir 4235	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐴) = ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴))
20		inidm 4175	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐴) = 𝐴
21		incom 4158	. . . . . . . . . . . 12 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
22	20, 21	uneq12i 4116	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴)) = (𝐴 ∪ (𝐴 ∩ 𝐵))
23		unabs 4214	. . . . . . . . . . 11 ⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴
24	22, 23	eqtri 2785	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐴) ∪ (𝐵 ∩ 𝐴)) = 𝐴
25	19, 24	eqtri 2785	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐴) = 𝐴
26	25	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∩ 𝐴) = 𝐴)
27	26	fveq2d 6886	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (vol*‘((𝐴 ∪ 𝐵) ∩ 𝐴)) = (vol*‘𝐴))
28		uncom 4108	. . . . . . . . . . 11 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
29	28	difeq1i 4073	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐴) = ((𝐵 ∪ 𝐴) ∖ 𝐴)
30		difun2 4440	. . . . . . . . . 10 ⊢ ((𝐵 ∪ 𝐴) ∖ 𝐴) = (𝐵 ∖ 𝐴)
31	29, 30	eqtri 2785	. . . . . . . . 9 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐴) = (𝐵 ∖ 𝐴)
32	21	eqeq1i 2767	. . . . . . . . . 10 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)
33		disj3 4410	. . . . . . . . . 10 ⊢ ((𝐵 ∩ 𝐴) = ∅ ↔ 𝐵 = (𝐵 ∖ 𝐴))
34	32, 33	sylbb1 240	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) = ∅ → 𝐵 = (𝐵 ∖ 𝐴))
35	31, 34	eqtr4id 2816	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐴) = 𝐵)
36	35	fveq2d 6886	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (vol*‘((𝐴 ∪ 𝐵) ∖ 𝐴)) = (vol*‘𝐵))
37	27, 36	oveq12d 7435	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((vol*‘((𝐴 ∪ 𝐵) ∩ 𝐴)) + (vol*‘((𝐴 ∪ 𝐵) ∖ 𝐴))) = ((vol*‘𝐴) + (vol*‘𝐵)))
38	18, 37	syl 18	. . . . 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 2797	. . . 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 418	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) → (((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘(𝐴 ∪ 𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
41		mblvol 25764	. . . . . 6 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
42	41	eleq1d 2847	. . . . 5 ⊢ (𝐴 ∈ dom vol → ((vol‘𝐴) ∈ ℝ ↔ (vol*‘𝐴) ∈ ℝ))
43		mblvol 25764	. . . . . 6 ⊢ (𝐵 ∈ dom vol → (vol‘𝐵) = (vol*‘𝐵))
44	43	eleq1d 2847	. . . . 5 ⊢ (𝐵 ∈ dom vol → ((vol‘𝐵) ∈ ℝ ↔ (vol*‘𝐵) ∈ ℝ))
45	42, 44	bi2anan9 650	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) ↔ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
46	45	3adant3 1150	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) → (((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ) ↔ ((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
47		unmbl 25771	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ∈ dom vol)
48		mblvol 25764	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ dom vol → (vol‘(𝐴 ∪ 𝐵)) = (vol*‘(𝐴 ∪ 𝐵)))
49	47, 48	syl 18	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (vol‘(𝐴 ∪ 𝐵)) = (vol*‘(𝐴 ∪ 𝐵)))
50	41, 43	oveqan12d 7436	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘𝐴) + (vol‘𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵)))
51	49, 50	eqeq12d 2778	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → ((vol‘(𝐴 ∪ 𝐵)) = ((vol‘𝐴) + (vol‘𝐵)) ↔ (vol*‘(𝐴 ∪ 𝐵)) = ((vol*‘𝐴) + (vol*‘𝐵))))
52	51	3adant3 1150	. . 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 412	1 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((vol‘𝐴) ∈ ℝ ∧ (vol‘𝐵) ∈ ℝ)) → (vol‘(𝐴 ∪ 𝐵)) = ((vol‘𝐴) + (vol‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282   class class class wbr 5107  dom cdm 5659  ‘cfv 6537  (class class class)co 7417  ℝcr 11127   + caddc 11131   ≤ cle 11272  vol*covol 25696  volcvol 25697

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-sup 9416  df-inf 9417  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-n0 12533  df-z 12620  df-uz 12892  df-q 13002  df-rp 13047  df-ioo 13406  df-ico 13408  df-icc 13409  df-fz 13566  df-fl 13857  df-seq 14070  df-exp 14130  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-ovol 25698  df-vol 25699

This theorem is used by:  volinun  25780  volfiniun  25781  volsup  25790  ovolioo  25802  ismblfin  38418  volioc  46808  volico  46819