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Theorem dpval 28587
Description: Define the value of the decimal point operator. See df-dp 28585. (Contributed by David A. Wheeler, 15-May-2015.)
Assertion
Ref Expression
dpval  |-  ( ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )

Proof of Theorem dpval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dp2 28584 . . 3  |- _ x y  =  ( x  +  ( y  /  10 ) )
2 oveq1 6091 . . 3  |-  ( x  =  A  ->  (
x  +  ( y  /  10 ) )  =  ( A  +  ( y  /  10 ) ) )
31, 2syl5eq 2482 . 2  |-  ( x  =  A  -> _ x y  =  ( A  +  ( y  /  10 ) ) )
4 oveq1 6091 . . . 4  |-  ( y  =  B  ->  (
y  /  10 )  =  ( B  /  10 ) )
54oveq2d 6100 . . 3  |-  ( y  =  B  ->  ( A  +  ( y  /  10 ) )  =  ( A  +  ( B  /  10 ) ) )
6 df-dp2 28584 . . 3  |- _ A B  =  ( A  +  ( B  /  10 ) )
75, 6syl6eqr 2488 . 2  |-  ( y  =  B  ->  ( A  +  ( y  /  10 ) )  = _ A B )
8 df-dp 28585 . 2  |-  period  =  ( x  e.  NN0 , 
y  e.  RR  |-> _ x y )
9 ovex 6109 . . 3  |-  ( A  +  ( B  /  10 ) )  e.  _V
106, 9eqeltri 2508 . 2  |- _ A B  e.  _V
113, 7, 8, 10ovmpt2 6212 1  |-  ( ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958  (class class class)co 6084   RRcr 8994    + caddc 8998    / cdiv 9682   10c10 10062   NN0cn0 10226  _cdp2 28582   periodcdp 28583
This theorem is referenced by:  dpcl  28588  dpfrac1  28589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-dp2 28584  df-dp 28585
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