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Theorem dpval 28231
Description: Define the value of the decimal point operator. See df-dp 28229. (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 28228 . . 3  |- _ x y  =  ( x  +  ( y  /  10 ) )
2 oveq1 6051 . . 3  |-  ( x  =  A  ->  (
x  +  ( y  /  10 ) )  =  ( A  +  ( y  /  10 ) ) )
31, 2syl5eq 2452 . 2  |-  ( x  =  A  -> _ x y  =  ( A  +  ( y  /  10 ) ) )
4 oveq1 6051 . . . 4  |-  ( y  =  B  ->  (
y  /  10 )  =  ( B  /  10 ) )
54oveq2d 6060 . . 3  |-  ( y  =  B  ->  ( A  +  ( y  /  10 ) )  =  ( A  +  ( B  /  10 ) ) )
6 df-dp2 28228 . . 3  |- _ A B  =  ( A  +  ( B  /  10 ) )
75, 6syl6eqr 2458 . 2  |-  ( y  =  B  ->  ( A  +  ( y  /  10 ) )  = _ A B )
8 df-dp 28229 . 2  |-  period  =  ( x  e.  NN0 , 
y  e.  RR  |-> _ x y )
9 ovex 6069 . . 3  |-  ( A  +  ( B  /  10 ) )  e.  _V
106, 9eqeltri 2478 . 2  |- _ A B  e.  _V
113, 7, 8, 10ovmpt2 6172 1  |-  ( ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2920  (class class class)co 6044   RRcr 8949    + caddc 8953    / cdiv 9637   10c10 10017   NN0cn0 10181  _cdp2 28226   periodcdp 28227
This theorem is referenced by:  dpcl  28232  dpfrac1  28233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-dp2 28228  df-dp 28229
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