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Theorem ddeval0 28670
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 9612 . . . . 5  |-  RR  e.  _V
21ssex 4537 . . . 4  |-  ( A 
C_  RR  ->  A  e. 
_V )
3 elpwg 3962 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  ~P RR  <->  A 
C_  RR ) )
43biimpar 483 . . . 4  |-  ( ( A  e.  _V  /\  A  C_  RR )  ->  A  e.  ~P RR )
52, 4mpancom 667 . . 3  |-  ( A 
C_  RR  ->  A  e. 
~P RR )
6 eleq2 2475 . . . . 5  |-  ( a  =  A  ->  (
0  e.  a  <->  0  e.  A ) )
76ifbid 3906 . . . 4  |-  ( a  =  A  ->  if ( 0  e.  a ,  1 ,  0 )  =  if ( 0  e.  A , 
1 ,  0 ) )
8 df-dde 28668 . . . 4  |- δ  =  ( a  e.  ~P RR  |->  if ( 0  e.  a ,  1 ,  0 ) )
9 1ex 9620 . . . . 5  |-  1  e.  _V
10 c0ex 9619 . . . . 5  |-  0  e.  _V
119, 10ifex 3952 . . . 4  |-  if ( 0  e.  A , 
1 ,  0 )  e.  _V
127, 8, 11fvmpt 5931 . . 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 3893 . 2  |-  ( -.  0  e.  A  ->  if ( 0  e.  A ,  1 ,  0 )  =  0 )
1513, 14sylan9eq 2463 1  |-  ( ( A  C_  RR  /\  -.  0  e.  A )  ->  (δ `  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    C_ wss 3413   ifcif 3884   ~Pcpw 3954   ` cfv 5568   RRcr 9520   0cc0 9521   1c1 9522  δcdde 28667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-mulcl 9583  ax-i2m1 9589
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-dde 28668
This theorem is referenced by:  ddemeas  28671
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