Theorem decbin2 9927

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

Step	Hyp	Ref	Expression
1		2t1e2 9461	. . 3 |- ( 2 x. 1 ) = 2
2	1	oveq2i 6096	. 2 |- ( ( 2 x. ( 2 x. A ) ) + ( 2 x. 1 ) ) = ( ( 2 x. ( 2 x. A ) ) + 2 )
3		2cn 9378	. . 3 |- 2 e. CC
4		decbin.1	. . . . 5 |- A e. NN0
5	4	nn0cni 9580	. . . 4 |- A e. CC
6	3, 5	mulcli 8332	. . 3 |- ( 2 x. A ) e. CC
7		ax-1cn 8273	. . 3 |- 1 e. CC
8	3, 6, 7	adddii 8337	. 2 |- ( 2 x. ( ( 2 x. A ) + 1 ) ) = ( ( 2 x. ( 2 x. A ) ) + ( 2 x. 1 ) )
9	4	decbin0 9926	. . 3 |- ( 4 x. A ) = ( 2 x. ( 2 x. A ) )
10	9	oveq1i 6095	. 2 |- ( ( 4 x. A ) + 2 ) = ( ( 2 x. ( 2 x. A ) ) + 2 )
11	2, 8, 10	3eqtr4ri 2270	1 |- ( ( 4 x. A ) + 2 ) = ( 2 x. ( ( 2 x. A ) + 1 ) )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6085   1c1 8181   + caddc 8183   x. cmul 8185   2c2 9358   4c4 9360   NN0cn0 9568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4249  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-1rid 8287  ax-rnegex 8289  ax-cnre 8291

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-n0 9569

This theorem is used by:  decbin3  9928