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Theorem measfrge0 26761
Description: A measure is a function over its base to the positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
measfrge0  |-  ( M  e.  (measures `  S
)  ->  M : S
--> ( 0 [,] +oo ) )

Proof of Theorem measfrge0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 measbase 26755 . . . 4  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
2 ismeas 26757 . . . 4  |-  ( 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 ) ) ) ) )
31, 2syl 16 . . 3  |-  ( 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 ) ) ) ) )
43ibi 241 . 2  |-  ( 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 ) ) ) )
54simp1d 1000 1  |-  ( M  e.  (measures `  S
)  ->  M : S
--> ( 0 [,] +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2798   (/)c0 3744   ~Pcpw 3967   U.cuni 4198  Disj wdisj 4369   class class class wbr 4399   ran crn 4948   -->wf 5521   ` cfv 5525  (class class class)co 6199   omcom 6585    ~<_ cdom 7417   0cc0 9392   +oocpnf 9525   [,]cicc 11413  Σ*cesum 26627  sigAlgebracsiga 26694  measurescmeas 26753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6202  df-esum 26628  df-meas 26754
This theorem is referenced by:  measfn  26762  measvxrge0  26763  meascnbl  26777  measres  26780  measdivcstOLD  26782  measdivcst  26783
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