Theorem bitsp1 12514

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 9305	. . . . . . . . . . . 12 |- 2 e. NN
2	1	a1i 9	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. NN0 ) -> 2 e. NN )
3	2	nncnd 9157	. . . . . . . . . 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 10937	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ ( M + 1 ) ) = ( ( 2 ^ M ) x. 2 ) )
6	2, 4	nnexpcld 10958	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ M ) e. NN )
7	6	nncnd 9157	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ M ) e. CC )
8	7, 3	mulcomd 8201	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( 2 ^ M ) x. 2 ) = ( 2 x. ( 2 ^ M ) ) )
9	5, 8	eqtrd 2264	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ ( M + 1 ) ) = ( 2 x. ( 2 ^ M ) ) )
10	9	oveq2d 6034	. . . . . . 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 9603	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> N e. CC )
13	2	nnap0d 9189	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> 2 # 0 )
14	6	nnap0d 9189	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( 2 ^ M ) # 0 )
15	12, 3, 7, 13, 14	divdivap1d 9002	. . . . . . 7 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( ( N / 2 ) / ( 2 ^ M ) ) = ( N / ( 2 x. ( 2 ^ M ) ) ) )
16	10, 15	eqtr4d 2267	. . . . . 6 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( N / ( 2 ^ ( M + 1 ) ) ) = ( ( N / 2 ) / ( 2 ^ M ) ) )
17	16	fveq2d 5643	. . . . 5 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) = ( |_ ` ( ( N / 2 ) / ( 2 ^ M ) ) ) )
18		znq 9858	. . . . . . 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 10584	. . . . . 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 2267	. . . 4 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( |_ ` ( N / ( 2 ^ ( M + 1 ) ) ) ) = ( |_ ` ( ( |_ ` ( N / 2 ) ) / ( 2 ^ M ) ) ) )
23	22	breq2d 4100	. . 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 9442	. . 3 |- ( M e. NN0 -> ( M + 1 ) e. NN0 )
26		bitsval2 12507	. . 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 10538	. . 3 |- ( ( N e. ZZ /\ M e. NN0 ) -> ( |_ ` ( N / 2 ) ) e. ZZ )
29		bitsval2 12507	. . 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 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   1c1 8033   + caddc 8035   x. cmul 8037   / cdiv 8852   NNcn 9143   2c2 9194   NN0cn0 9402   ZZcz 9479   QQcq 9853   |_cfl 10529   ^cexp 10801   || cdvds 12350  bitscbits 12503

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-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  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-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  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-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  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-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fl 10531  df-seqfrec 10711  df-exp 10802  df-bits 12504

This theorem is referenced by:  bitsp1e  12515  bitsp1o  12516