Theorem measval 30261

Description: The value of the measures function applied on a sigma-algebra. (Contributed by Thierry Arnoux, 17-Oct-2016.)

Assertion

Ref	Expression
measval	|- ( S e. U. ran sigAlgebra -> (measures ` S ) = { 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 ) ) ) } )

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

Allowed substitution hint:   S( y)

Proof of Theorem measval

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( 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 ) )
2	1	ss2abi 3674	. . 3 |- { 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 ) }
3		ovex 6678	. . . 4 |- ( 0 [,] +oo ) e. _V
4		mapex 7863	. . . 4 |- ( ( S e. U. ran sigAlgebra /\ ( 0 [,] +oo ) e. _V ) -> { m | m : S --> ( 0 [,] +oo ) } e. _V )
5	3, 4	mpan2 707	. . 3 |- ( S e. U. ran sigAlgebra -> { m | m : S --> ( 0 [,] +oo ) } e. _V )
6		ssexg 4804	. . 3 |- ( ( { 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 ) } /\ { m | m : S --> ( 0 [,] +oo ) } e. _V ) -> { 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 )
7	2, 5, 6	sylancr 695	. 2 |- ( 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 ) ) ) } e. _V )
8		feq2 6027	. . . . 5 |- ( s = S -> ( m : s --> ( 0 [,] +oo ) <-> m : S --> ( 0 [,] +oo ) ) )
9		pweq 4161	. . . . . 6 |- ( s = S -> ~P s = ~P S )
10	9	raleqdv 3144	. . . . 5 |- ( s = S -> ( 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 ) ) ) )
11	8, 10	3anbi13d 1401	. . . 4 |- ( s = 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 ) ) ) <-> ( 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 ) ) ) ) )
12	11	abbidv 2741	. . 3 |- ( s = S -> { 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 | ( 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 ) ) ) } )
13		df-meas 30259	. . 3 |- 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 ) ) ) } )
14	12, 13	fvmptg 6280	. 2 |- ( ( 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 ) ) ) } e. _V ) -> (measures ` S ) = { 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 ) ) ) } )
15	7, 14	mpdan 702	1 |- ( S e. U. ran sigAlgebra -> (measures ` S ) = { 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 ) ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)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-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-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-meas 30259

This theorem is referenced by: (None)