Theorem bitsval 12509

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 12507	. . . 4 |- bits = ( n e. ZZ |-> { m e. NN0 | -. 2 || ( |_ ` ( n / ( 2 ^ m ) ) ) } )
2	1	mptrcl 5729	. . 3 |- ( M e. (bits ` N ) -> N e. ZZ )
3		bitsfval 12508	. . . . 5 |- ( N e. ZZ -> (bits ` N ) = { m e. NN0 | -. 2 || ( |_ ` ( N / ( 2 ^ m ) ) ) } )
4	3	eleq2d 2301	. . . 4 |- ( N e. ZZ -> ( M e. (bits ` N ) <-> M e. { m e. NN0 | -. 2 || ( |_ ` ( N / ( 2 ^ m ) ) ) } ) )
5		oveq2 6026	. . . . . . . . 9 |- ( m = M -> ( 2 ^ m ) = ( 2 ^ M ) )
6	5	oveq2d 6034	. . . . . . . 8 |- ( m = M -> ( N / ( 2 ^ m ) ) = ( N / ( 2 ^ M ) ) )
7	6	fveq2d 5643	. . . . . . 7 |- ( m = M -> ( |_ ` ( N / ( 2 ^ m ) ) ) = ( |_ ` ( N / ( 2 ^ M ) ) ) )
8	7	breq2d 4100	. . . . . 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 2962	. . . 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 1008	. 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 1004   = wceq 1397   e. wcel 2202   {crab 2514   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   / cdiv 8852   2c2 9194   NN0cn0 9402   ZZcz 9479   |_cfl 10529   ^cexp 10801   || cdvds 12353  bitscbits 12506

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-i2m1 8137

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6021  df-inn 9144  df-n0 9403  df-bits 12507

This theorem is referenced by:  bitsval2  12510  bitsss  12511  bitsfzo  12521  bitsmod  12522  bitscmp  12524