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Theorem ddeval0 26603
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 9365 . . . . 5  |-  RR  e.  _V
21ssex 4431 . . . 4  |-  ( A 
C_  RR  ->  A  e. 
_V )
3 elpwg 3863 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  ~P RR  <->  A 
C_  RR ) )
43biimpar 485 . . . 4  |-  ( ( A  e.  _V  /\  A  C_  RR )  ->  A  e.  ~P RR )
52, 4mpancom 669 . . 3  |-  ( A 
C_  RR  ->  A  e. 
~P RR )
6 eleq2 2499 . . . . 5  |-  ( a  =  A  ->  (
0  e.  a  <->  0  e.  A ) )
76ifbid 3806 . . . 4  |-  ( a  =  A  ->  if ( 0  e.  a ,  1 ,  0 )  =  if ( 0  e.  A , 
1 ,  0 ) )
8 df-dde 26601 . . . 4  |- δ  =  ( a  e.  ~P RR  |->  if ( 0  e.  a ,  1 ,  0 ) )
9 1ex 9373 . . . . 5  |-  1  e.  _V
10 c0ex 9372 . . . . 5  |-  0  e.  _V
119, 10ifex 3853 . . . 4  |-  if ( 0  e.  A , 
1 ,  0 )  e.  _V
127, 8, 11fvmpt 5769 . . 3  |-  ( A  e.  ~P RR  ->  (δ `  A )  =  if ( 0  e.  A ,  1 ,  0 ) )
135, 12syl 16 . 2  |-  ( A 
C_  RR  ->  (δ `  A
)  =  if ( 0  e.  A , 
1 ,  0 ) )
14 iffalse 3794 . 2  |-  ( -.  0  e.  A  ->  if ( 0  e.  A ,  1 ,  0 )  =  0 )
1513, 14sylan9eq 2490 1  |-  ( ( A  C_  RR  /\  -.  0  e.  A )  ->  (δ `  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967    C_ wss 3323   ifcif 3786   ~Pcpw 3855   ` cfv 5413   RRcr 9273   0cc0 9274   1c1 9275  δcdde 26600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-mulcl 9336  ax-i2m1 9342
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-dde 26601
This theorem is referenced by:  ddemeas  26604
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