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Theorem ddeval0 29058
Description: Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
Assertion
Ref Expression
ddeval0  |-  ( ( A  C_  RR  /\  -.  0  e.  A )  ->  (δ `  A )  =  0 )

Proof of Theorem ddeval0
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 reex 9630 . . . . 5  |-  RR  e.  _V
21ssex 4547 . . . 4  |-  ( A 
C_  RR  ->  A  e. 
_V )
3 elpwg 3959 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  ~P RR  <->  A 
C_  RR ) )
43biimpar 488 . . . 4  |-  ( ( A  e.  _V  /\  A  C_  RR )  ->  A  e.  ~P RR )
52, 4mpancom 675 . . 3  |-  ( A 
C_  RR  ->  A  e. 
~P RR )
6 eleq2 2518 . . . . 5  |-  ( a  =  A  ->  (
0  e.  a  <->  0  e.  A ) )
76ifbid 3903 . . . 4  |-  ( a  =  A  ->  if ( 0  e.  a ,  1 ,  0 )  =  if ( 0  e.  A , 
1 ,  0 ) )
8 df-dde 29056 . . . 4  |- δ  =  ( a  e.  ~P RR  |->  if ( 0  e.  a ,  1 ,  0 ) )
9 1ex 9638 . . . . 5  |-  1  e.  _V
10 c0ex 9637 . . . . 5  |-  0  e.  _V
119, 10ifex 3949 . . . 4  |-  if ( 0  e.  A , 
1 ,  0 )  e.  _V
127, 8, 11fvmpt 5948 . . 3  |-  ( A  e.  ~P RR  ->  (δ `  A )  =  if ( 0  e.  A ,  1 ,  0 ) )
135, 12syl 17 . 2  |-  ( A 
C_  RR  ->  (δ `  A
)  =  if ( 0  e.  A , 
1 ,  0 ) )
14 iffalse 3890 . 2  |-  ( -.  0  e.  A  ->  if ( 0  e.  A ,  1 ,  0 )  =  0 )
1513, 14sylan9eq 2505 1  |-  ( ( A  C_  RR  /\  -.  0  e.  A )  ->  (δ `  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045    C_ wss 3404   ifcif 3881   ~Pcpw 3951   ` cfv 5582   RRcr 9538   0cc0 9539   1c1 9540  δcdde 29055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-mulcl 9601  ax-i2m1 9607
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-dde 29056
This theorem is referenced by:  ddemeas  29059
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