Theorem dpval 29597

Description: Define the value of the decimal point operator. See df-dp 29596. (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.

Step	Hyp	Ref	Expression
1		df-dp2 29578	. . 3 |- _ x y = ( x + ( y / ; 1 0 ) )
2		oveq1 6657	. . 3 |- ( x = A -> ( x + ( y / ; 1 0 ) ) = ( A + ( y / ; 1 0 ) ) )
3	1, 2	syl5eq 2668	. 2 |- ( x = A -> _ x y = ( A + ( y / ; 1 0 ) ) )
4		oveq1 6657	. . . 4 |- ( y = B -> ( y / ; 1 0 ) = ( B / ; 1 0 ) )
5	4	oveq2d 6666	. . 3 |- ( y = B -> ( A + ( y / ; 1 0 ) ) = ( A + ( B / ; 1 0 ) ) )
6		df-dp2 29578	. . 3 |- _ A B = ( A + ( B / ; 1 0 ) )
7	5, 6	syl6eqr 2674	. 2 |- ( y = B -> ( A + ( y / ; 1 0 ) ) = _ A B )
8		df-dp 29596	. 2 |- period = ( x e. NN0 , y e. RR |-> _ x y )
9	6	ovexi 6679	. 2 |- _ A B e. _V
10	3, 7, 8, 9	ovmpt2 6796	1 |- ( ( A e. NN0 /\ B e. RR ) -> ( A period B ) = _ A B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   / cdiv 10684   NN0cn0 11292  ;cdc 11493  _cdp2 29577   periodcdp 29595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-dp2 29578  df-dp 29596

This theorem is referenced by:  dpcl  29598  dpfrac1  29599  dpfrac1OLD  29600  dpval2  29601  dpmul1000  29607  dpadd2  29618