Theorem ismeas 30262

Description: The property of being a measure. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 19-Oct-2016.)

Assertion

Ref	Expression
ismeas	|- ( S e. U. ran sigAlgebra -> ( M e. (measures ` S ) <-> ( M : S --> ( 0 [,] +oo ) /\ ( M ` (/) ) = 0 /\ A. x e. ~P S ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = Σ* y e. x ( M ` y ) ) ) ) )

Distinct variable groups:   x, y, M   x, S

Allowed substitution hint:   S( y)

Proof of Theorem ismeas

Dummy variables m s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( M e. (measures ` S ) -> M e. _V )
2	1	a1i 11	. 2 |- ( S e. U. ran sigAlgebra -> ( M e. (measures ` S ) -> M e. _V ) )
3		simp1 1061	. . 3 |- ( ( M : S --> ( 0 [,] +oo ) /\ ( M ` (/) ) = 0 /\ A. x e. ~P S ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = Σ* y e. x ( M ` y ) ) ) -> M : S --> ( 0 [,] +oo ) )
4		ovex 6678	. . . 4 |- ( 0 [,] +oo ) e. _V
5		fex2 7121	. . . . . 6 |- ( ( M : S --> ( 0 [,] +oo ) /\ S e. U. ran sigAlgebra /\ ( 0 [,] +oo ) e. _V ) -> M e. _V )
6	5	3expb 1266	. . . . 5 |- ( ( M : S --> ( 0 [,] +oo ) /\ ( S e. U. ran sigAlgebra /\ ( 0 [,] +oo ) e. _V ) ) -> M e. _V )
7	6	expcom 451	. . . 4 |- ( ( S e. U. ran sigAlgebra /\ ( 0 [,] +oo ) e. _V ) -> ( M : S --> ( 0 [,] +oo ) -> M e. _V ) )
8	4, 7	mpan2 707	. . 3 |- ( S e. U. ran sigAlgebra -> ( M : S --> ( 0 [,] +oo ) -> M e. _V ) )
9	3, 8	syl5 34	. 2 |- ( S e. U. ran sigAlgebra -> ( ( M : S --> ( 0 [,] +oo ) /\ ( M ` (/) ) = 0 /\ A. x e. ~P S ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = Σ* y e. x ( M ` y ) ) ) -> M e. _V ) )
10		df-meas 30259	. . . 4 |- measures = ( s e. U. ran sigAlgebra |-> { m | ( m : s --> ( 0 [,] +oo ) /\ ( m ` (/) ) = 0 /\ A. x e. ~P s ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) ) } )
11		vex 3203	. . . . . 6 |- s e. _V
12		mapex 7863	. . . . . 6 |- ( ( s e. _V /\ ( 0 [,] +oo ) e. _V ) -> { m | m : s --> ( 0 [,] +oo ) } e. _V )
13	11, 4, 12	mp2an 708	. . . . 5 |- { m | m : s --> ( 0 [,] +oo ) } e. _V
14		simp1 1061	. . . . . 6 |- ( ( m : s --> ( 0 [,] +oo ) /\ ( m ` (/) ) = 0 /\ A. x e. ~P s ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) ) -> m : s --> ( 0 [,] +oo ) )
15	14	ss2abi 3674	. . . . 5 |- { m | ( m : s --> ( 0 [,] +oo ) /\ ( m ` (/) ) = 0 /\ A. x e. ~P s ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) ) } C_ { m | m : s --> ( 0 [,] +oo ) }
16	13, 15	ssexi 4803	. . . 4 |- { m | ( m : s --> ( 0 [,] +oo ) /\ ( m ` (/) ) = 0 /\ A. x e. ~P s ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) ) } e. _V
17		simpr 477	. . . . . 6 |- ( ( s = S /\ m = M ) -> m = M )
18		simpl 473	. . . . . 6 |- ( ( s = S /\ m = M ) -> s = S )
19	17, 18	feq12d 6033	. . . . 5 |- ( ( s = S /\ m = M ) -> ( m : s --> ( 0 [,] +oo ) <-> M : S --> ( 0 [,] +oo ) ) )
20		fveq1 6190	. . . . . . 7 |- ( m = M -> ( m ` (/) ) = ( M ` (/) ) )
21	20	eqeq1d 2624	. . . . . 6 |- ( m = M -> ( ( m ` (/) ) = 0 <-> ( M ` (/) ) = 0 ) )
22	21	adantl 482	. . . . 5 |- ( ( s = S /\ m = M ) -> ( ( m ` (/) ) = 0 <-> ( M ` (/) ) = 0 ) )
23	18	pweqd 4163	. . . . . 6 |- ( ( s = S /\ m = M ) -> ~P s = ~P S )
24		fveq1 6190	. . . . . . . . 9 |- ( m = M -> ( m ` U. x ) = ( M ` U. x ) )
25		fveq1 6190	. . . . . . . . . 10 |- ( m = M -> ( m ` y ) = ( M ` y ) )
26	25	esumeq2sdv 30101	. . . . . . . . 9 |- ( m = M -> Σ* y e. x ( m ` y ) = Σ* y e. x ( M ` y ) )
27	24, 26	eqeq12d 2637	. . . . . . . 8 |- ( m = M -> ( ( m ` U. x ) = Σ* y e. x ( m ` y ) <-> ( M ` U. x ) = Σ* y e. x ( M ` y ) ) )
28	27	imbi2d 330	. . . . . . 7 |- ( m = M -> ( ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) <-> ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = Σ* y e. x ( M ` y ) ) ) )
29	28	adantl 482	. . . . . 6 |- ( ( s = S /\ m = M ) -> ( ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) <-> ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = Σ* y e. x ( M ` y ) ) ) )
30	23, 29	raleqbidv 3152	. . . . 5 |- ( ( s = S /\ m = M ) -> ( A. x e. ~P s ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) <-> A. x e. ~P S ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = Σ* y e. x ( M ` y ) ) ) )
31	19, 22, 30	3anbi123d 1399	. . . 4 |- ( ( s = S /\ m = M ) -> ( ( m : s --> ( 0 [,] +oo ) /\ ( m ` (/) ) = 0 /\ A. x e. ~P s ( ( x ~<_ om /\ Disj y e. x y ) -> ( m ` U. x ) = Σ* y e. x ( m ` y ) ) ) <-> ( M : S --> ( 0 [,] +oo ) /\ ( M ` (/) ) = 0 /\ A. x e. ~P S ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = Σ* y e. x ( M ` y ) ) ) ) )
32	10, 16, 31	abfmpel 29455	. . 3 |- ( ( S e. U. ran sigAlgebra /\ M e. _V ) -> ( M e. (measures ` S ) <-> ( M : S --> ( 0 [,] +oo ) /\ ( M ` (/) ) = 0 /\ A. x e. ~P S ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = Σ* y e. x ( M ` y ) ) ) ) )
33	32	ex 450	. 2 |- ( S e. U. ran sigAlgebra -> ( M e. _V -> ( M e. (measures ` S ) <-> ( M : S --> ( 0 [,] +oo ) /\ ( M ` (/) ) = 0 /\ A. x e. ~P S ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = Σ* y e. x ( M ` y ) ) ) ) ) )
34	2, 9, 33	pm5.21ndd 369	1 |- ( S e. U. ran sigAlgebra -> ( M e. (measures ` S ) <-> ( M : S --> ( 0 [,] +oo ) /\ ( M ` (/) ) = 0 /\ A. x e. ~P S ( ( x ~<_ om /\ Disj y e. x y ) -> ( M ` U. x ) = Σ* y e. x ( M ` y ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   (/)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-ral 2917  df-rex 2918  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-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:  measbasedom  30265  measfrge0  30266  measvnul  30269  measvun  30272  measinb  30284  measres  30285  measdivcstOLD  30287  measdivcst  30288  cntmeas  30289  volmeas  30294  ddemeas  30299  omsmeas  30385  dstrvprob  30533