Theorem measfrge0 30266

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.

Step	Hyp	Ref	Expression
1		measbase 30260	. . . 4 |- ( M e. (measures ` S ) -> S e. U. ran sigAlgebra )
2		ismeas 30262	. . . 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 ) ) ) ) )
3	1, 2	syl 17	. . 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 ) ) ) ) )
4	3	ibi 256	. 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 ) ) ) )
5	4	simp1d 1073	1 |- ( M e. (measures ` S ) -> M : S --> ( 0 [,] +oo ) )

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-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:  measfn  30267  measvxrge0  30268  meascnbl  30282  measres  30285  measdivcstOLD  30287  measdivcst  30288