Theorem bitsp1 12428

Description: The M + 1-th bit of N is the M-th bit of |_ ( N / 2 ). (Contributed by Mario Carneiro, 5-Sep-2016.)

Assertion

Ref	Expression
bitsp1	|- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( M + 1 ) e. (bits ` N ) <-> M e. (bits ` ( |_ ` ( N / 2 ) ) ) ) )

Proof of Theorem bitsp1

Step	Hyp	Ref	Expression
1		2nn 9240	. . . . . . . . . . . 12 |- 2 e. NN
2	1	a1i 9	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. NN0 ) -> 2 e. NN )
3	2	nncnd 9092	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. NN0 ) -> 2 e. CC )
4		simpr 110	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. NN0 ) -> M e. NN0 )
5	3, 4	expp1d 10863	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ ( M + 1 ) ) = ( ( 2 ^ M ) x. 2 ) )
6	2, 4	nnexpcld 10884	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ M ) e. NN )
7	6	nncnd 9092	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ M ) e. CC )
8	7, 3	mulcomd 8136	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( 2 ^ M ) x. 2 ) = ( 2 x. ( 2 ^ M ) ) )
9	5, 8	eqtrd 2242	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ ( M + 1 ) ) = ( 2 x. ( 2 ^ M ) ) )
10	9	oveq2d 5990	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( N / ( 2 ^ ( M + 1 ) ) ) = ( N / ( 2 x. ( 2 ^ M ) ) ) )
11		simpl 109	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. NN0 ) -> N e. ZZ )
12	11	zcnd 9538	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> N e. CC )
13	2	nnap0d 9124	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> 2 # 0 )
14	6	nnap0d 9124	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ M ) # 0 )
15	12, 3, 7, 13, 14	divdivap1d 8937	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( N / 2 ) / ( 2 ^ M ) ) = ( N / ( 2 x. ( 2 ^ M ) ) ) )
16	10, 15	eqtr4d 2245	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( N / ( 2 ^ ( M + 1 ) ) ) = ( ( N / 2 ) / ( 2 ^ M ) ) )
17	16	fveq2d 5607	. . . . 5 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) = ( |_ ` ( ( N / 2 ) / ( 2 ^ M ) ) ) )
18		znq 9787	. . . . . . 7 |- ( ( N e. ZZ /\ 2 e. NN ) -> ( N / 2 ) e. QQ )
19	2, 18	syldan 282	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( N / 2 ) e. QQ )
20		flqdiv 10510	. . . . . 6 |- ( ( ( N / 2 ) e. QQ /\ ( 2 ^ M ) e. NN ) -> ( |_ ` ( ( |_ ` ( N / 2 ) ) / ( 2 ^ M ) ) ) = ( |_ ` ( ( N / 2 ) / ( 2 ^ M ) ) ) )
21	19, 6, 20	syl2anc 411	. . . . 5 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( |_ ` ( ( |_ ` ( N / 2 ) ) / ( 2 ^ M ) ) ) = ( |_ ` ( ( N / 2 ) / ( 2 ^ M ) ) ) )
22	17, 21	eqtr4d 2245	. . . 4 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) = ( |_ ` ( ( |_ ` ( N / 2 ) ) / ( 2 ^ M ) ) ) )
23	22	breq2d 4074	. . 3 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 || ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) <-> 2 || ( |_ ` ( ( |_ ` ( N / 2 ) ) / ( 2 ^ M ) ) ) ) )
24	23	notbid 671	. 2 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( -. 2 || ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) <-> -. 2 || ( |_ ` ( ( |_ ` ( N / 2 ) ) / ( 2 ^ M ) ) ) ) )
25		peano2nn0 9377	. . 3 |- ( M e. NN0 -> ( M + 1 ) e. NN0 )
26		bitsval2 12421	. . 3 |- ( ( N e. ZZ /\ ( M + 1 ) e. NN0 ) -> ( ( M + 1 ) e. (bits ` N ) <-> -. 2 || ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) ) )
27	25, 26	sylan2 286	. 2 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( M + 1 ) e. (bits ` N ) <-> -. 2 || ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) ) )
28	19	flqcld 10464	. . 3 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( |_ ` ( N / 2 ) ) e. ZZ )
29		bitsval2 12421	. . 3 |- ( ( ( |_ ` ( N / 2 ) ) e. ZZ /\ M e. NN0 ) -> ( M e. (bits ` ( |_ ` ( N / 2 ) ) ) <-> -. 2 || ( |_ ` ( ( |_ ` ( N / 2 ) ) / ( 2 ^ M ) ) ) ) )
30	28, 29	sylancom 420	. 2 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( M e. (bits ` ( |_ ` ( N / 2 ) ) ) <-> -. 2 || ( |_ ` ( ( |_ ` ( N / 2 ) ) / ( 2 ^ M ) ) ) ) )
31	24, 27, 30	3bitr4d 220	1 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( M + 1 ) e. (bits ` N ) <-> M e. (bits ` ( |_ ` ( N / 2 ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1375   e. wcel 2180   class class class wbr 4062   ` cfv 5294  (class class class)co 5974   1c1 7968   + caddc 7970   x. cmul 7972   / cdiv 8787   NNcn 9078   2c2 9129   NN0cn0 9337   ZZcz 9414   QQcq 9782   |_cfl 10455   ^cexp 10727   || cdvds 12264  bitscbits 12417

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-n0 9338  df-z 9415  df-uz 9691  df-q 9783  df-rp 9818  df-fl 10457  df-seqfrec 10637  df-exp 10728  df-bits 12418

This theorem is referenced by:  bitsp1e  12429  bitsp1o  12430