Theorem decbin2 9867

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 9408	. . 3 |- ( 2 x. 1 ) = 2
2	1	oveq2i 6069	. 2 |- ( ( 2 x. ( 2 x. A ) ) + ( 2 x. 1 ) ) = ( ( 2 x. ( 2 x. A ) ) + 2 )
3		2cn 9325	. . 3 |- 2 e. CC
4		decbin.1	. . . . 5 |- A e. NN0
5	4	nn0cni 9525	. . . 4 |- A e. CC
6	3, 5	mulcli 8295	. . 3 |- ( 2 x. A ) e. CC
7		ax-1cn 8236	. . 3 |- 1 e. CC
8	3, 6, 7	adddii 8300	. 2 |- ( 2 x. ( ( 2 x. A ) + 1 ) ) = ( ( 2 x. ( 2 x. A ) ) + ( 2 x. 1 ) )
9	4	decbin0 9866	. . 3 |- ( 4 x. A ) = ( 2 x. ( 2 x. A ) )
10	9	oveq1i 6068	. 2 |- ( ( 4 x. A ) + 2 ) = ( ( 2 x. ( 2 x. A ) ) + 2 )
11	2, 8, 10	3eqtr4ri 2266	1 |- ( ( 4 x. A ) + 2 ) = ( 2 x. ( ( 2 x. A ) + 1 ) )

Syntax hints:   = wceq 1398   e. wcel 2205  (class class class)co 6058   1c1 8144   + caddc 8146   x. cmul 8148   2c2 9305   4c4 9307   NN0cn0 9513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-sep 4233  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-1rid 8250  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514

This theorem is referenced by:  decbin3  9868