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Theorem ddeval1 26770
Description: Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
Assertion
Ref Expression
ddeval1  |-  ( ( A  C_  RR  /\  0  e.  A )  ->  (δ `  A )  =  1 )

Proof of Theorem ddeval1
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 reex 9460 . . . . 5  |-  RR  e.  _V
21ssex 4520 . . . 4  |-  ( A 
C_  RR  ->  A  e. 
_V )
3 elpwg 3952 . . . . 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 2521 . . . . 5  |-  ( a  =  A  ->  (
0  e.  a  <->  0  e.  A ) )
76ifbid 3895 . . . 4  |-  ( a  =  A  ->  if ( 0  e.  a ,  1 ,  0 )  =  if ( 0  e.  A , 
1 ,  0 ) )
8 df-dde 26769 . . . 4  |- δ  =  ( a  e.  ~P RR  |->  if ( 0  e.  a ,  1 ,  0 ) )
9 1ex 9468 . . . . 5  |-  1  e.  _V
10 c0ex 9467 . . . . 5  |-  0  e.  _V
119, 10ifex 3942 . . . 4  |-  if ( 0  e.  A , 
1 ,  0 )  e.  _V
127, 8, 11fvmpt 5859 . . 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 iftrue 3881 . 2  |-  ( 0  e.  A  ->  if ( 0  e.  A ,  1 ,  0 )  =  1 )
1513, 14sylan9eq 2510 1  |-  ( ( A  C_  RR  /\  0  e.  A )  ->  (δ `  A )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757   _Vcvv 3054    C_ wss 3412   ifcif 3875   ~Pcpw 3944   ` cfv 5502   RRcr 9368   0cc0 9369   1c1 9370  δcdde 26768
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 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-mulcl 9431  ax-i2m1 9437
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-iota 5465  df-fun 5504  df-fv 5510  df-dde 26769
This theorem is referenced by:  ddemeas  26772
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