Theorem bitsuz 15196

Description: The bits of a number are all at least N iff the number is divisible by 2 ^ N. (Contributed by Mario Carneiro, 21-Sep-2016.)

Assertion

Ref	Expression
bitsuz	|- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( 2 ^ N ) || A <-> (bits ` A ) C_ ( ZZ>= ` N ) ) )

Proof of Theorem bitsuz

Step	Hyp	Ref	Expression
1		bitsres 15195	. . . 4 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` A ) i^i ( ZZ>= ` N ) ) = (bits ` ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) ) )
2	1	eqeq1d 2624	. . 3 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( (bits ` A ) i^i ( ZZ>= ` N ) ) = (bits ` A ) <-> (bits ` ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) ) = (bits ` A ) ) )
3		simpl 473	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN0 ) -> A e. ZZ )
4	3	zred 11482	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN0 ) -> A e. RR )
5		2nn 11185	. . . . . . . . 9 |- 2 e. NN
6	5	a1i 11	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN0 ) -> 2 e. NN )
7		simpr 477	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN0 ) -> N e. NN0 )
8	6, 7	nnexpcld 13030	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( 2 ^ N ) e. NN )
9	4, 8	nndivred 11069	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( A / ( 2 ^ N ) ) e. RR )
10	9	flcld 12599	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( |_ ` ( A / ( 2 ^ N ) ) ) e. ZZ )
11	8	nnzd 11481	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( 2 ^ N ) e. ZZ )
12	10, 11	zmulcld 11488	. . . 4 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) e. ZZ )
13		bitsf1 15168	. . . . 5 |- bits : ZZ -1-1-> ~P NN0
14		f1fveq 6519	. . . . 5 |- ( (bits : ZZ -1-1-> ~P NN0 /\ ( ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) e. ZZ /\ A e. ZZ ) ) -> ( (bits ` ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) ) = (bits ` A ) <-> ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) = A ) )
15	13, 14	mpan 706	. . . 4 |- ( ( ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) e. ZZ /\ A e. ZZ ) -> ( (bits ` ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) ) = (bits ` A ) <-> ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) = A ) )
16	12, 3, 15	syl2anc 693	. . 3 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) ) = (bits ` A ) <-> ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) = A ) )
17		dvdsmul2 15004	. . . . . 6 |- ( ( ( |_ ` ( A / ( 2 ^ N ) ) ) e. ZZ /\ ( 2 ^ N ) e. ZZ ) -> ( 2 ^ N ) || ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) )
18	10, 11, 17	syl2anc 693	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( 2 ^ N ) || ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) )
19		breq2 4657	. . . . 5 |- ( ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) = A -> ( ( 2 ^ N ) || ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) <-> ( 2 ^ N ) || A ) )
20	18, 19	syl5ibcom 235	. . . 4 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) = A -> ( 2 ^ N ) || A ) )
21	8	nnne0d 11065	. . . . . . . . . 10 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( 2 ^ N ) =/= 0 )
22		dvdsval2 14986	. . . . . . . . . 10 |- ( ( ( 2 ^ N ) e. ZZ /\ ( 2 ^ N ) =/= 0 /\ A e. ZZ ) -> ( ( 2 ^ N ) || A <-> ( A / ( 2 ^ N ) ) e. ZZ ) )
23	11, 21, 3, 22	syl3anc 1326	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( 2 ^ N ) || A <-> ( A / ( 2 ^ N ) ) e. ZZ ) )
24	23	biimpa 501	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. NN0 ) /\ ( 2 ^ N ) || A ) -> ( A / ( 2 ^ N ) ) e. ZZ )
25		flid 12609	. . . . . . . 8 |- ( ( A / ( 2 ^ N ) ) e. ZZ -> ( |_ ` ( A / ( 2 ^ N ) ) ) = ( A / ( 2 ^ N ) ) )
26	24, 25	syl 17	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. NN0 ) /\ ( 2 ^ N ) || A ) -> ( |_ ` ( A / ( 2 ^ N ) ) ) = ( A / ( 2 ^ N ) ) )
27	26	oveq1d 6665	. . . . . 6 |- ( ( ( A e. ZZ /\ N e. NN0 ) /\ ( 2 ^ N ) || A ) -> ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) = ( ( A / ( 2 ^ N ) ) x. ( 2 ^ N ) ) )
28	3	zcnd 11483	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN0 ) -> A e. CC )
29	28	adantr 481	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. NN0 ) /\ ( 2 ^ N ) || A ) -> A e. CC )
30	8	nncnd 11036	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( 2 ^ N ) e. CC )
31	30	adantr 481	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. NN0 ) /\ ( 2 ^ N ) || A ) -> ( 2 ^ N ) e. CC )
32		2cnd 11093	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. NN0 ) /\ ( 2 ^ N ) || A ) -> 2 e. CC )
33		2ne0 11113	. . . . . . . . 9 |- 2 =/= 0
34	33	a1i 11	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. NN0 ) /\ ( 2 ^ N ) || A ) -> 2 =/= 0 )
35	7	nn0zd 11480	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. NN0 ) -> N e. ZZ )
36	35	adantr 481	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. NN0 ) /\ ( 2 ^ N ) || A ) -> N e. ZZ )
37	32, 34, 36	expne0d 13014	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. NN0 ) /\ ( 2 ^ N ) || A ) -> ( 2 ^ N ) =/= 0 )
38	29, 31, 37	divcan1d 10802	. . . . . 6 |- ( ( ( A e. ZZ /\ N e. NN0 ) /\ ( 2 ^ N ) || A ) -> ( ( A / ( 2 ^ N ) ) x. ( 2 ^ N ) ) = A )
39	27, 38	eqtrd 2656	. . . . 5 |- ( ( ( A e. ZZ /\ N e. NN0 ) /\ ( 2 ^ N ) || A ) -> ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) = A )
40	39	ex 450	. . . 4 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( 2 ^ N ) || A -> ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) = A ) )
41	20, 40	impbid 202	. . 3 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) = A <-> ( 2 ^ N ) || A ) )
42	2, 16, 41	3bitrrd 295	. 2 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( 2 ^ N ) || A <-> ( (bits ` A ) i^i ( ZZ>= ` N ) ) = (bits ` A ) ) )
43		df-ss 3588	. 2 |- ( (bits ` A ) C_ ( ZZ>= ` N ) <-> ( (bits ` A ) i^i ( ZZ>= ` N ) ) = (bits ` A ) )
44	42, 43	syl6bbr 278	1 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( 2 ^ N ) || A <-> (bits ` A ) C_ ( ZZ>= ` N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   |_cfl 12591   ^cexp 12860   || cdvds 14983  bitscbits 15141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  df-tru 1486  df-fal 1489  df-had 1533  df-cad 1546  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-bits 15144  df-sad 15173

This theorem is referenced by:  bitsshft  15197