Theorem bitsp1 12630

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 9395	. . . . . . . . . . . 12 |- 2 e. NN
2	1	a1i 9	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. NN0 ) -> 2 e. NN )
3	2	nncnd 9247	. . . . . . . . . 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 11032	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ ( M + 1 ) ) = ( ( 2 ^ M ) x. 2 ) )
6	2, 4	nnexpcld 11053	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ M ) e. NN )
7	6	nncnd 9247	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ M ) e. CC )
8	7, 3	mulcomd 8291	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( 2 ^ M ) x. 2 ) = ( 2 x. ( 2 ^ M ) ) )
9	5, 8	eqtrd 2265	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ ( M + 1 ) ) = ( 2 x. ( 2 ^ M ) ) )
10	9	oveq2d 6065	. . . . . . 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 9697	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> N e. CC )
13	2	nnap0d 9279	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> 2 # 0 )
14	6	nnap0d 9279	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ M ) # 0 )
15	12, 3, 7, 13, 14	divdivap1d 9092	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( N / 2 ) / ( 2 ^ M ) ) = ( N / ( 2 x. ( 2 ^ M ) ) ) )
16	10, 15	eqtr4d 2268	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( N / ( 2 ^ ( M + 1 ) ) ) = ( ( N / 2 ) / ( 2 ^ M ) ) )
17	16	fveq2d 5673	. . . . 5 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) = ( |_ ` ( ( N / 2 ) / ( 2 ^ M ) ) ) )
18		znq 9952	. . . . . . 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 10679	. . . . . 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 2268	. . . 4 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) = ( |_ ` ( ( |_ ` ( N / 2 ) ) / ( 2 ^ M ) ) ) )
23	22	breq2d 4120	. . 3 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 || ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) <-> 2 || ( |_ ` ( ( |_ ` ( N / 2 ) ) / ( 2 ^ M ) ) ) ) )
24	23	notbid 673	. 2 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( -. 2 || ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) <-> -. 2 || ( |_ ` ( ( |_ ` ( N / 2 ) ) / ( 2 ^ M ) ) ) ) )
25		peano2nn0 9532	. . 3 |- ( M e. NN0 -> ( M + 1 ) e. NN0 )
26		bitsval2 12623	. . 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 10633	. . 3 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( |_ ` ( N / 2 ) ) e. ZZ )
29		bitsval2 12623	. . 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 1398   e. wcel 2203   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   1c1 8124   + caddc 8126   x. cmul 8128   / cdiv 8942   NNcn 9233   2c2 9284   NN0cn0 9492   ZZcz 9573   QQcq 9947   |_cfl 10624   ^cexp 10896   || cdvds 12466  bitscbits 12619

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-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  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-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  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-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-n0 9493  df-z 9574  df-uz 9850  df-q 9948  df-rp 9983  df-fl 10626  df-seqfrec 10806  df-exp 10897  df-bits 12620

This theorem is referenced by:  bitsp1e  12631  bitsp1o  12632