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Theorem dyadval 18947
Description: Value of the dyadic rational function  F. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypothesis
Ref Expression
dyadmbl.1  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
Assertion
Ref Expression
dyadval  |-  ( ( A  e.  ZZ  /\  B  e.  NN0 )  -> 
( A F B )  =  <. ( A  /  ( 2 ^ B ) ) ,  ( ( A  + 
1 )  /  (
2 ^ B ) ) >. )
Distinct variable groups:    x, y, B    x, A, y    x, F, y

Proof of Theorem dyadval
StepHypRef Expression
1 id 19 . . . 4  |-  ( x  =  A  ->  x  =  A )
2 oveq2 5866 . . . 4  |-  ( y  =  B  ->  (
2 ^ y )  =  ( 2 ^ B ) )
31, 2oveqan12d 5877 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  /  (
2 ^ y ) )  =  ( A  /  ( 2 ^ B ) ) )
4 oveq1 5865 . . . 4  |-  ( x  =  A  ->  (
x  +  1 )  =  ( A  + 
1 ) )
54, 2oveqan12d 5877 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  + 
1 )  /  (
2 ^ y ) )  =  ( ( A  +  1 )  /  ( 2 ^ B ) ) )
63, 5opeq12d 3804 . 2  |-  ( ( x  =  A  /\  y  =  B )  -> 
<. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.  =  <. ( A  / 
( 2 ^ B
) ) ,  ( ( A  +  1 )  /  ( 2 ^ B ) )
>. )
7 dyadmbl.1 . 2  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
8 opex 4237 . 2  |-  <. ( A  /  ( 2 ^ B ) ) ,  ( ( A  + 
1 )  /  (
2 ^ B ) ) >.  e.  _V
96, 7, 8ovmpt2a 5978 1  |-  ( ( A  e.  ZZ  /\  B  e.  NN0 )  -> 
( A F B )  =  <. ( A  /  ( 2 ^ B ) ) ,  ( ( A  + 
1 )  /  (
2 ^ B ) ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643  (class class class)co 5858    e. cmpt2 5860   1c1 8738    + caddc 8740    / cdiv 9423   2c2 9795   NN0cn0 9965   ZZcz 10024   ^cexp 11104
This theorem is referenced by:  dyadovol  18948  dyadss  18949  dyaddisjlem  18950  dyadmaxlem  18952  opnmbllem  18956
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
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