Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ddeval0 Structured version   Unicode version

Theorem ddeval0 26815
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 9487 . . . . 5  |-  RR  e.  _V
21ssex 4547 . . . 4  |-  ( A 
C_  RR  ->  A  e. 
_V )
3 elpwg 3979 . . . . 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 2527 . . . . 5  |-  ( a  =  A  ->  (
0  e.  a  <->  0  e.  A ) )
76ifbid 3922 . . . 4  |-  ( a  =  A  ->  if ( 0  e.  a ,  1 ,  0 )  =  if ( 0  e.  A , 
1 ,  0 ) )
8 df-dde 26813 . . . 4  |- δ  =  ( a  e.  ~P RR  |->  if ( 0  e.  a ,  1 ,  0 ) )
9 1ex 9495 . . . . 5  |-  1  e.  _V
10 c0ex 9494 . . . . 5  |-  0  e.  _V
119, 10ifex 3969 . . . 4  |-  if ( 0  e.  A , 
1 ,  0 )  e.  _V
127, 8, 11fvmpt 5886 . . 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 3910 . 2  |-  ( -.  0  e.  A  ->  if ( 0  e.  A ,  1 ,  0 )  =  0 )
1513, 14sylan9eq 2515 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 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3439   ifcif 3902   ~Pcpw 3971   ` cfv 5529   RRcr 9395   0cc0 9396   1c1 9397  δcdde 26812
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-mulcl 9458  ax-i2m1 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  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 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-dde 26813
This theorem is referenced by:  ddemeas  26816
  Copyright terms: Public domain W3C validator