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Theorem decbin0 8733
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
decbin0  |-  ( 4  x.  A )  =  ( 2  x.  (
2  x.  A ) )

Proof of Theorem decbin0
StepHypRef Expression
1 2t2e4 8289 . . 3  |-  ( 2  x.  2 )  =  4
21oveq1i 5574 . 2  |-  ( ( 2  x.  2 )  x.  A )  =  ( 4  x.  A
)
3 2cn 8213 . . 3  |-  2  e.  CC
4 decbin.1 . . . 4  |-  A  e. 
NN0
54nn0cni 8403 . . 3  |-  A  e.  CC
63, 3, 5mulassi 7226 . 2  |-  ( ( 2  x.  2 )  x.  A )  =  ( 2  x.  (
2  x.  A ) )
72, 6eqtr3i 2105 1  |-  ( 4  x.  A )  =  ( 2  x.  (
2  x.  A ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434  (class class class)co 5564    x. cmul 7084   2c2 8192   4c4 8194   NN0cn0 8391
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 2065  ax-sep 3917  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-mulcom 7175  ax-addass 7176  ax-mulass 7177  ax-distr 7178  ax-1rid 7181  ax-rnegex 7183  ax-cnre 7185
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-br 3807  df-iota 4918  df-fv 4961  df-ov 5567  df-inn 8143  df-2 8201  df-3 8202  df-4 8203  df-n0 8392
This theorem is referenced by:  decbin2  8734
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