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Theorem dpval 32462
Description: Define the value of the decimal point operator. See df-dp 32460. (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 32459 . . 3  |- _ x y  =  ( x  +  ( y  /  10 ) )
2 oveq1 6292 . . 3  |-  ( x  =  A  ->  (
x  +  ( y  /  10 ) )  =  ( A  +  ( y  /  10 ) ) )
31, 2syl5eq 2520 . 2  |-  ( x  =  A  -> _ x y  =  ( A  +  ( y  /  10 ) ) )
4 oveq1 6292 . . . 4  |-  ( y  =  B  ->  (
y  /  10 )  =  ( B  /  10 ) )
54oveq2d 6301 . . 3  |-  ( y  =  B  ->  ( A  +  ( y  /  10 ) )  =  ( A  +  ( B  /  10 ) ) )
6 df-dp2 32459 . . 3  |- _ A B  =  ( A  +  ( B  /  10 ) )
75, 6syl6eqr 2526 . 2  |-  ( y  =  B  ->  ( A  +  ( y  /  10 ) )  = _ A B )
8 df-dp 32460 . 2  |-  period  =  ( x  e.  NN0 , 
y  e.  RR  |-> _ x y )
9 ovex 6310 . . 3  |-  ( A  +  ( B  /  10 ) )  e.  _V
106, 9eqeltri 2551 . 2  |- _ A B  e.  _V
113, 7, 8, 10ovmpt2 6423 1  |-  ( ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113  (class class class)co 6285   RRcr 9492    + caddc 9496    / cdiv 10207   10c10 10594   NN0cn0 10796  _cdp2 32457   periodcdp 32458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-dp2 32459  df-dp 32460
This theorem is referenced by:  dpcl  32463  dpfrac1  32464
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