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Theorem dpval 31105
Description: Define the value of the decimal point operator. See df-dp 31103. (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 31102 . . 3  |- _ x y  =  ( x  +  ( y  /  10 ) )
2 oveq1 6098 . . 3  |-  ( x  =  A  ->  (
x  +  ( y  /  10 ) )  =  ( A  +  ( y  /  10 ) ) )
31, 2syl5eq 2487 . 2  |-  ( x  =  A  -> _ x y  =  ( A  +  ( y  /  10 ) ) )
4 oveq1 6098 . . . 4  |-  ( y  =  B  ->  (
y  /  10 )  =  ( B  /  10 ) )
54oveq2d 6107 . . 3  |-  ( y  =  B  ->  ( A  +  ( y  /  10 ) )  =  ( A  +  ( B  /  10 ) ) )
6 df-dp2 31102 . . 3  |- _ A B  =  ( A  +  ( B  /  10 ) )
75, 6syl6eqr 2493 . 2  |-  ( y  =  B  ->  ( A  +  ( y  /  10 ) )  = _ A B )
8 df-dp 31103 . 2  |-  period  =  ( x  e.  NN0 , 
y  e.  RR  |-> _ x y )
9 ovex 6116 . . 3  |-  ( A  +  ( B  /  10 ) )  e.  _V
106, 9eqeltri 2513 . 2  |- _ A B  e.  _V
113, 7, 8, 10ovmpt2 6226 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 1369    e. wcel 1756   _Vcvv 2972  (class class class)co 6091   RRcr 9281    + caddc 9285    / cdiv 9993   10c10 10379   NN0cn0 10579  _cdp2 31100   periodcdp 31101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-dp2 31102  df-dp 31103
This theorem is referenced by:  dpcl  31106  dpfrac1  31107
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