Theorem measvun 30272

Description: The measure of a countable disjoint union is the sum of the measures. (Contributed by Thierry Arnoux, 26-Dec-2016.)

Assertion

Ref	Expression
measvun	|- ( ( M e. (measures ` S ) /\ A e. ~P S /\ ( A ~<_ om /\ Disj x e. A x ) ) -> ( M ` U. A ) = Σ* x e. A ( M ` x ) )

Distinct variable groups:   x, A   x, M

Allowed substitution hint:   S( x)

Proof of Theorem measvun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. 2 |- ( ( M e. (measures ` S ) /\ A e. ~P S /\ ( A ~<_ om /\ Disj x e. A x ) ) -> A e. ~P S )
2		measbase 30260	. . . . . 6 |- ( M e. (measures ` S ) -> S e. U. ran sigAlgebra )
3		ismeas 30262	. . . . . 6 |- ( S e. U. ran sigAlgebra -> ( M e. (measures ` S ) <-> ( M : S --> ( 0 [,] +oo ) /\ ( M ` (/) ) = 0 /\ A. y e. ~P S ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = Σ* x e. y ( M ` x ) ) ) ) )
4	2, 3	syl 17	. . . . 5 |- ( M e. (measures ` S ) -> ( M e. (measures ` S ) <-> ( M : S --> ( 0 [,] +oo ) /\ ( M ` (/) ) = 0 /\ A. y e. ~P S ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = Σ* x e. y ( M ` x ) ) ) ) )
5	4	ibi 256	. . . 4 |- ( M e. (measures ` S ) -> ( M : S --> ( 0 [,] +oo ) /\ ( M ` (/) ) = 0 /\ A. y e. ~P S ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = Σ* x e. y ( M ` x ) ) ) )
6	5	simp3d 1075	. . 3 |- ( M e. (measures ` S ) -> A. y e. ~P S ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = Σ* x e. y ( M ` x ) ) )
7	6	3ad2ant1 1082	. 2 |- ( ( M e. (measures ` S ) /\ A e. ~P S /\ ( A ~<_ om /\ Disj x e. A x ) ) -> A. y e. ~P S ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = Σ* x e. y ( M ` x ) ) )
8		simp3 1063	. 2 |- ( ( M e. (measures ` S ) /\ A e. ~P S /\ ( A ~<_ om /\ Disj x e. A x ) ) -> ( A ~<_ om /\ Disj x e. A x ) )
9		breq1 4656	. . . . 5 |- ( y = A -> ( y ~<_ om <-> A ~<_ om ) )
10		disjeq1 4627	. . . . 5 |- ( y = A -> (Disj x e. y x <-> Disj x e. A x ) )
11	9, 10	anbi12d 747	. . . 4 |- ( y = A -> ( ( y ~<_ om /\ Disj x e. y x ) <-> ( A ~<_ om /\ Disj x e. A x ) ) )
12		unieq 4444	. . . . . 6 |- ( y = A -> U. y = U. A )
13	12	fveq2d 6195	. . . . 5 |- ( y = A -> ( M ` U. y ) = ( M ` U. A ) )
14		esumeq1 30096	. . . . 5 |- ( y = A -> Σ* x e. y ( M ` x ) = Σ* x e. A ( M ` x ) )
15	13, 14	eqeq12d 2637	. . . 4 |- ( y = A -> ( ( M ` U. y ) = Σ* x e. y ( M ` x ) <-> ( M ` U. A ) = Σ* x e. A ( M ` x ) ) )
16	11, 15	imbi12d 334	. . 3 |- ( y = A -> ( ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = Σ* x e. y ( M ` x ) ) <-> ( ( A ~<_ om /\ Disj x e. A x ) -> ( M ` U. A ) = Σ* x e. A ( M ` x ) ) ) )
17	16	rspcv 3305	. 2 |- ( A e. ~P S -> ( A. y e. ~P S ( ( y ~<_ om /\ Disj x e. y x ) -> ( M ` U. y ) = Σ* x e. y ( M ` x ) ) -> ( ( A ~<_ om /\ Disj x e. A x ) -> ( M ` U. A ) = Σ* x e. A ( M ` x ) ) ) )
18	1, 7, 8, 17	syl3c 66	1 |- ( ( M e. (measures ` S ) /\ A e. ~P S /\ ( A ~<_ om /\ Disj x e. A x ) ) -> ( M ` U. A ) = Σ* x e. A ( M ` x ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   ~Pcpw 4158   U.cuni 4436  Disj wdisj 4620   class class class wbr 4653   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   0cc0 9936   +oocpnf 10071   [,]cicc 12178  Σ^*cesum 30089  sigAlgebracsiga 30170  measurescmeas 30258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-esum 30090  df-meas 30259

This theorem is referenced by:  measxun2  30273  measvunilem  30275  measssd  30278  measres  30285  measdivcstOLD  30287  measdivcst  30288  probcun  30480  totprobd  30488