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Theorem dyadval 18963
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 5882 . . . 4  |-  ( y  =  B  ->  (
2 ^ y )  =  ( 2 ^ B ) )
31, 2oveqan12d 5893 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  /  (
2 ^ y ) )  =  ( A  /  ( 2 ^ B ) ) )
4 oveq1 5881 . . . 4  |-  ( x  =  A  ->  (
x  +  1 )  =  ( A  + 
1 ) )
54, 2oveqan12d 5893 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  + 
1 )  /  (
2 ^ y ) )  =  ( ( A  +  1 )  /  ( 2 ^ B ) ) )
63, 5opeq12d 3820 . 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 4253 . 2  |-  <. ( A  /  ( 2 ^ B ) ) ,  ( ( A  + 
1 )  /  (
2 ^ B ) ) >.  e.  _V
96, 7, 8ovmpt2a 5994 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 1632    e. wcel 1696   <.cop 3656  (class class class)co 5874    e. cmpt2 5876   1c1 8754    + caddc 8756    / cdiv 9439   2c2 9811   NN0cn0 9981   ZZcz 10040   ^cexp 11120
This theorem is referenced by:  dyadovol  18964  dyadss  18965  dyaddisjlem  18966  dyadmaxlem  18968  opnmbllem  18972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879
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