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Theorem dyadval 19472
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 20 . . . 4  |-  ( x  =  A  ->  x  =  A )
2 oveq2 6080 . . . 4  |-  ( y  =  B  ->  (
2 ^ y )  =  ( 2 ^ B ) )
31, 2oveqan12d 6091 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  /  (
2 ^ y ) )  =  ( A  /  ( 2 ^ B ) ) )
4 oveq1 6079 . . . 4  |-  ( x  =  A  ->  (
x  +  1 )  =  ( A  + 
1 ) )
54, 2oveqan12d 6091 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  + 
1 )  /  (
2 ^ y ) )  =  ( ( A  +  1 )  /  ( 2 ^ B ) ) )
63, 5opeq12d 3984 . 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 4419 . 2  |-  <. ( A  /  ( 2 ^ B ) ) ,  ( ( A  + 
1 )  /  (
2 ^ B ) ) >.  e.  _V
96, 7, 8ovmpt2a 6195 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 359    = wceq 1652    e. wcel 1725   <.cop 3809  (class class class)co 6072    e. cmpt2 6074   1c1 8980    + caddc 8982    / cdiv 9666   2c2 10038   NN0cn0 10210   ZZcz 10271   ^cexp 11370
This theorem is referenced by:  dyadovol  19473  dyadss  19474  dyaddisjlem  19475  dyadmaxlem  19477  opnmbllem  19481  mblfinlem  26190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077
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