Theorem bitsval 12625

Description: Expand the definition of the bits of an integer. (Contributed by Mario Carneiro, 5-Sep-2016.)

Assertion

Ref	Expression
bitsval	|- ( M e. (bits ` N ) <-> ( N e. ZZ /\ M e. NN0 /\ -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) )

Proof of Theorem bitsval

Dummy variables n m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bits 12623	. . . 4 |- bits = ( n e. ZZ |-> { m e. NN0 | -. 2 || ( |_ ` ( n / ( 2 ^ m ) ) ) } )
2	1	mptrcl 5759	. . 3 |- ( M e. (bits ` N ) -> N e. ZZ )
3		bitsfval 12624	. . . . 5 |- ( N e. ZZ -> (bits ` N ) = { m e. NN0 | -. 2 || ( |_ ` ( N / ( 2 ^ m ) ) ) } )
4	3	eleq2d 2302	. . . 4 |- ( N e. ZZ -> ( M e. (bits ` N ) <-> M e. { m e. NN0 | -. 2 || ( |_ ` ( N / ( 2 ^ m ) ) ) } ) )
5		oveq2 6057	. . . . . . . . 9 |- ( m = M -> ( 2 ^ m ) = ( 2 ^ M ) )
6	5	oveq2d 6065	. . . . . . . 8 |- ( m = M -> ( N / ( 2 ^ m ) ) = ( N / ( 2 ^ M ) ) )
7	6	fveq2d 5673	. . . . . . 7 |- ( m = M -> ( |_ ` ( N / ( 2 ^ m ) ) ) = ( |_ ` ( N / ( 2 ^ M ) ) ) )
8	7	breq2d 4120	. . . . . 6 |- ( m = M -> ( 2 || ( |_ ` ( N / ( 2 ^ m ) ) ) <-> 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) )
9	8	notbid 673	. . . . 5 |- ( m = M -> ( -. 2 || ( |_ ` ( N / ( 2 ^ m ) ) ) <-> -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) )
10	9	elrab 2972	. . . 4 |- ( M e. { m e. NN0 | -. 2 || ( |_ ` ( N / ( 2 ^ m ) ) ) } <-> ( M e. NN0 /\ -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) )
11	4, 10	bitrdi 196	. . 3 |- ( N e. ZZ -> ( M e. (bits ` N ) <-> ( M e. NN0 /\ -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) ) )
12	2, 11	biadanii 617	. 2 |- ( M e. (bits ` N ) <-> ( N e. ZZ /\ ( M e. NN0 /\ -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) ) )
13		3anass 1009	. 2 |- ( ( N e. ZZ /\ M e. NN0 /\ -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) <-> ( N e. ZZ /\ ( M e. NN0 /\ -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) ) )
14	12, 13	bitr4i 187	1 |- ( M e. (bits ` N ) <-> ( N e. ZZ /\ M e. NN0 /\ -. 2 || ( |_ ` ( N / ( 2 ^ M ) ) ) ) )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   {crab 2524   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   / cdiv 8945   2c2 9287   NN0cn0 9495   ZZcz 9576   |_cfl 10627   ^cexp 10899   || cdvds 12469  bitscbits 12622

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-in1 619  ax-in2 620  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-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-i2m1 8231

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fv 5359  df-ov 6052  df-inn 9237  df-n0 9496  df-bits 12623

This theorem is referenced by:  bitsval2  12626  bitsss  12627  bitsfzo  12637  bitsmod  12638  bitscmp  12640