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Theorem dyadval 19489
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 21 . . . 4  |-  ( x  =  A  ->  x  =  A )
2 oveq2 6092 . . . 4  |-  ( y  =  B  ->  (
2 ^ y )  =  ( 2 ^ B ) )
31, 2oveqan12d 6103 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  /  (
2 ^ y ) )  =  ( A  /  ( 2 ^ B ) ) )
4 oveq1 6091 . . . 4  |-  ( x  =  A  ->  (
x  +  1 )  =  ( A  + 
1 ) )
54, 2oveqan12d 6103 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  + 
1 )  /  (
2 ^ y ) )  =  ( ( A  +  1 )  /  ( 2 ^ B ) ) )
63, 5opeq12d 3994 . 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 4430 . 2  |-  <. ( A  /  ( 2 ^ B ) ) ,  ( ( A  + 
1 )  /  (
2 ^ B ) ) >.  e.  _V
96, 7, 8ovmpt2a 6207 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 360    = wceq 1653    e. wcel 1726   <.cop 3819  (class class class)co 6084    e. cmpt2 6086   1c1 8996    + caddc 8998    / cdiv 9682   2c2 10054   NN0cn0 10226   ZZcz 10287   ^cexp 11387
This theorem is referenced by:  dyadovol  19490  dyadss  19491  dyaddisjlem  19492  dyadmaxlem  19494  opnmbllem  19498  opnmbllem0  26254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089
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