Theorem bitsval 12633

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 12631	. . . 4 |- bits = ( n e. ZZ |-> { m e. NN0 | -. 2 || ( |_ ` ( n / ( 2 ^ m ) ) ) } )
2	1	mptrcl 5762	. . 3 |- ( M e. (bits ` N ) -> N e. ZZ )
3		bitsfval 12632	. . . . 5 |- ( N e. ZZ -> (bits ` N ) = { m e. NN0 | -. 2 || ( |_ ` ( N / ( 2 ^ m ) ) ) } )
4	3	eleq2d 2304	. . . 4 |- ( N e. ZZ -> ( M e. (bits ` N ) <-> M e. { m e. NN0 | -. 2 || ( |_ ` ( N / ( 2 ^ m ) ) ) } ) )
5		oveq2 6060	. . . . . . . . 9 |- ( m = M -> ( 2 ^ m ) = ( 2 ^ M ) )
6	5	oveq2d 6068	. . . . . . . 8 |- ( m = M -> ( N / ( 2 ^ m ) ) = ( N / ( 2 ^ M ) ) )
7	6	fveq2d 5676	. . . . . . 7 |- ( m = M -> ( |_ ` ( N / ( 2 ^ m ) ) ) = ( |_ ` ( N / ( 2 ^ M ) ) ) )
8	7	breq2d 4123	. . . . . 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 2975	. . . 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 2205   {crab 2526   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   / cdiv 8948   2c2 9290   NN0cn0 9498   ZZcz 9579   |_cfl 10632   ^cexp 10904   || cdvds 12477  bitscbits 12630

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-i2m1 8234

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fv 5362  df-ov 6055  df-inn 9240  df-n0 9499  df-bits 12631

This theorem is referenced by:  bitsval2  12634  bitsss  12635  bitsfzo  12645  bitsmod  12646  bitscmp  12648