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Theorem decbin2 8698
Description: Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
Hypothesis
Ref Expression
decbin.1  |-  A  e. 
NN0
Assertion
Ref Expression
decbin2  |-  ( ( 4  x.  A )  +  2 )  =  ( 2  x.  (
( 2  x.  A
)  +  1 ) )

Proof of Theorem decbin2
StepHypRef Expression
1 2t1e2 8252 . . 3  |-  ( 2  x.  1 )  =  2
21oveq2i 5554 . 2  |-  ( ( 2  x.  ( 2  x.  A ) )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  ( 2  x.  A
) )  +  2 )
3 2cn 8177 . . 3  |-  2  e.  CC
4 decbin.1 . . . . 5  |-  A  e. 
NN0
54nn0cni 8367 . . . 4  |-  A  e.  CC
63, 5mulcli 7186 . . 3  |-  ( 2  x.  A )  e.  CC
7 ax-1cn 7131 . . 3  |-  1  e.  CC
83, 6, 7adddii 7191 . 2  |-  ( 2  x.  ( ( 2  x.  A )  +  1 ) )  =  ( ( 2  x.  ( 2  x.  A
) )  +  ( 2  x.  1 ) )
94decbin0 8697 . . 3  |-  ( 4  x.  A )  =  ( 2  x.  (
2  x.  A ) )
109oveq1i 5553 . 2  |-  ( ( 4  x.  A )  +  2 )  =  ( ( 2  x.  ( 2  x.  A
) )  +  2 )
112, 8, 103eqtr4ri 2113 1  |-  ( ( 4  x.  A )  +  2 )  =  ( 2  x.  (
( 2  x.  A
)  +  1 ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434  (class class class)co 5543   1c1 7044    + caddc 7046    x. cmul 7048   2c2 8156   4c4 8158   NN0cn0 8355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-1rid 7145  ax-rnegex 7147  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546  df-inn 8107  df-2 8165  df-3 8166  df-4 8167  df-n0 8356
This theorem is referenced by:  decbin3  8699
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