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Theorem dpval 28240
Description: Define the value of the decimal point operator. See df-dp 28238. (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 28237 . . 3  |- _ x y  =  ( x  +  ( y  /  10 ) )
2 oveq1 5865 . . 3  |-  ( x  =  A  ->  (
x  +  ( y  /  10 ) )  =  ( A  +  ( y  /  10 ) ) )
31, 2syl5eq 2327 . 2  |-  ( x  =  A  -> _ x y  =  ( A  +  ( y  /  10 ) ) )
4 oveq1 5865 . . . 4  |-  ( y  =  B  ->  (
y  /  10 )  =  ( B  /  10 ) )
54oveq2d 5874 . . 3  |-  ( y  =  B  ->  ( A  +  ( y  /  10 ) )  =  ( A  +  ( B  /  10 ) ) )
6 df-dp2 28237 . . 3  |- _ A B  =  ( A  +  ( B  /  10 ) )
75, 6syl6eqr 2333 . 2  |-  ( y  =  B  ->  ( A  +  ( y  /  10 ) )  = _ A B )
8 df-dp 28238 . 2  |-  period  =  ( x  e.  NN0 , 
y  e.  RR  |-> _ x y )
9 ovex 5883 . . 3  |-  ( A  +  ( B  /  10 ) )  e.  _V
106, 9eqeltri 2353 . 2  |- _ A B  e.  _V
113, 7, 8, 10ovmpt2 5983 1  |-  ( ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788  (class class class)co 5858   RRcr 8736    + caddc 8740    / cdiv 9423   10c10 9803   NN0cn0 9965  _cdp2 28235   periodcdp 28236
This theorem is referenced by:  dpcl  28241  dpfrac1  28242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-dp2 28237  df-dp 28238
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