Theorem bitsres 15195

Description: Restrict the bits of a number to an upper integer set. (Contributed by Mario Carneiro, 5-Sep-2016.)

Assertion

Ref	Expression
bitsres	|- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` A ) i^i ( ZZ>= ` N ) ) = (bits ` ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) ) )

Proof of Theorem bitsres

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN0 ) -> A e. ZZ )
2		2nn 11185	. . . . . . . 8 |- 2 e. NN
3	2	a1i 11	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN0 ) -> 2 e. NN )
4		simpr 477	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN0 ) -> N e. NN0 )
5	3, 4	nnexpcld 13030	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( 2 ^ N ) e. NN )
6	1, 5	zmodcld 12691	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( A mod ( 2 ^ N ) ) e. NN0 )
7	6	nn0zd 11480	. . . 4 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( A mod ( 2 ^ N ) ) e. ZZ )
8	7	znegcld 11484	. . 3 |- ( ( A e. ZZ /\ N e. NN0 ) -> -u ( A mod ( 2 ^ N ) ) e. ZZ )
9		sadadd 15189	. . 3 |- ( ( -u ( A mod ( 2 ^ N ) ) e. ZZ /\ A e. ZZ ) -> ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` A ) ) = (bits ` ( -u ( A mod ( 2 ^ N ) ) + A ) ) )
10	8, 1, 9	syl2anc 693	. 2 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` A ) ) = (bits ` ( -u ( A mod ( 2 ^ N ) ) + A ) ) )
11		sadadd 15189	. . . . . 6 |- ( ( -u ( A mod ( 2 ^ N ) ) e. ZZ /\ ( A mod ( 2 ^ N ) ) e. ZZ ) -> ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` ( A mod ( 2 ^ N ) ) ) ) = (bits ` ( -u ( A mod ( 2 ^ N ) ) + ( A mod ( 2 ^ N ) ) ) ) )
12	8, 7, 11	syl2anc 693	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` ( A mod ( 2 ^ N ) ) ) ) = (bits ` ( -u ( A mod ( 2 ^ N ) ) + ( A mod ( 2 ^ N ) ) ) ) )
13	8	zcnd 11483	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. NN0 ) -> -u ( A mod ( 2 ^ N ) ) e. CC )
14	7	zcnd 11483	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( A mod ( 2 ^ N ) ) e. CC )
15	13, 14	addcomd 10238	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( -u ( A mod ( 2 ^ N ) ) + ( A mod ( 2 ^ N ) ) ) = ( ( A mod ( 2 ^ N ) ) + -u ( A mod ( 2 ^ N ) ) ) )
16	14	negidd 10382	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( A mod ( 2 ^ N ) ) + -u ( A mod ( 2 ^ N ) ) ) = 0 )
17	15, 16	eqtrd 2656	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( -u ( A mod ( 2 ^ N ) ) + ( A mod ( 2 ^ N ) ) ) = 0 )
18	17	fveq2d 6195	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN0 ) -> (bits ` ( -u ( A mod ( 2 ^ N ) ) + ( A mod ( 2 ^ N ) ) ) ) = (bits ` 0 ) )
19		0bits 15161	. . . . . 6 |- (bits ` 0 ) = (/)
20	18, 19	syl6eq 2672	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> (bits ` ( -u ( A mod ( 2 ^ N ) ) + ( A mod ( 2 ^ N ) ) ) ) = (/) )
21	12, 20	eqtrd 2656	. . . 4 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` ( A mod ( 2 ^ N ) ) ) ) = (/) )
22	21	oveq1d 6665	. . 3 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` ( A mod ( 2 ^ N ) ) ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = ( (/) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) )
23		bitsss 15148	. . . . 5 |- (bits ` -u ( A mod ( 2 ^ N ) ) ) C_ NN0
24		bitsss 15148	. . . . 5 |- (bits ` ( A mod ( 2 ^ N ) ) ) C_ NN0
25		inss1 3833	. . . . . 6 |- ( (bits ` A ) i^i ( ZZ>= ` N ) ) C_ (bits ` A )
26		bitsss 15148	. . . . . . 7 |- (bits ` A ) C_ NN0
27	26	a1i 11	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN0 ) -> (bits ` A ) C_ NN0 )
28	25, 27	syl5ss 3614	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` A ) i^i ( ZZ>= ` N ) ) C_ NN0 )
29		sadass 15193	. . . . 5 |- ( ( (bits ` -u ( A mod ( 2 ^ N ) ) ) C_ NN0 /\ (bits ` ( A mod ( 2 ^ N ) ) ) C_ NN0 /\ ( (bits ` A ) i^i ( ZZ>= ` N ) ) C_ NN0 ) -> ( ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` ( A mod ( 2 ^ N ) ) ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd ( (bits ` ( A mod ( 2 ^ N ) ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) ) )
30	23, 24, 28, 29	mp3an12i 1428	. . . 4 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` ( A mod ( 2 ^ N ) ) ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd ( (bits ` ( A mod ( 2 ^ N ) ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) ) )
31		bitsmod 15158	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN0 ) -> (bits ` ( A mod ( 2 ^ N ) ) ) = ( (bits ` A ) i^i ( 0..^ N ) ) )
32	31	oveq1d 6665	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` ( A mod ( 2 ^ N ) ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = ( ( (bits ` A ) i^i ( 0..^ N ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) )
33		inss1 3833	. . . . . . . . . 10 |- ( (bits ` A ) i^i ( 0..^ N ) ) C_ (bits ` A )
34	33, 27	syl5ss 3614	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` A ) i^i ( 0..^ N ) ) C_ NN0 )
35		fzouzdisj 12504	. . . . . . . . . . . 12 |- ( ( 0..^ N ) i^i ( ZZ>= ` N ) ) = (/)
36	35	ineq2i 3811	. . . . . . . . . . 11 |- ( (bits ` A ) i^i ( ( 0..^ N ) i^i ( ZZ>= ` N ) ) ) = ( (bits ` A ) i^i (/) )
37		inindi 3830	. . . . . . . . . . 11 |- ( (bits ` A ) i^i ( ( 0..^ N ) i^i ( ZZ>= ` N ) ) ) = ( ( (bits ` A ) i^i ( 0..^ N ) ) i^i ( (bits ` A ) i^i ( ZZ>= ` N ) ) )
38		in0 3968	. . . . . . . . . . 11 |- ( (bits ` A ) i^i (/) ) = (/)
39	36, 37, 38	3eqtr3i 2652	. . . . . . . . . 10 |- ( ( (bits ` A ) i^i ( 0..^ N ) ) i^i ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = (/)
40	39	a1i 11	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( (bits ` A ) i^i ( 0..^ N ) ) i^i ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = (/) )
41	34, 28, 40	saddisj 15187	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( (bits ` A ) i^i ( 0..^ N ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = ( ( (bits ` A ) i^i ( 0..^ N ) ) u. ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) )
42		indi 3873	. . . . . . . 8 |- ( (bits ` A ) i^i ( ( 0..^ N ) u. ( ZZ>= ` N ) ) ) = ( ( (bits ` A ) i^i ( 0..^ N ) ) u. ( (bits ` A ) i^i ( ZZ>= ` N ) ) )
43	41, 42	syl6eqr 2674	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( (bits ` A ) i^i ( 0..^ N ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = ( (bits ` A ) i^i ( ( 0..^ N ) u. ( ZZ>= ` N ) ) ) )
44		nn0uz 11722	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
45	4, 44	syl6eleq 2711	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ N e. NN0 ) -> N e. ( ZZ>= ` 0 ) )
46		fzouzsplit 12503	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 0 ) -> ( ZZ>= ` 0 ) = ( ( 0..^ N ) u. ( ZZ>= ` N ) ) )
47	45, 46	syl 17	. . . . . . . . . 10 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ZZ>= ` 0 ) = ( ( 0..^ N ) u. ( ZZ>= ` N ) ) )
48	44, 47	syl5eq 2668	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. NN0 ) -> NN0 = ( ( 0..^ N ) u. ( ZZ>= ` N ) ) )
49	26, 48	syl5sseq 3653	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN0 ) -> (bits ` A ) C_ ( ( 0..^ N ) u. ( ZZ>= ` N ) ) )
50		df-ss 3588	. . . . . . . 8 |- ( (bits ` A ) C_ ( ( 0..^ N ) u. ( ZZ>= ` N ) ) <-> ( (bits ` A ) i^i ( ( 0..^ N ) u. ( ZZ>= ` N ) ) ) = (bits ` A ) )
51	49, 50	sylib 208	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` A ) i^i ( ( 0..^ N ) u. ( ZZ>= ` N ) ) ) = (bits ` A ) )
52	43, 51	eqtrd 2656	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( (bits ` A ) i^i ( 0..^ N ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = (bits ` A ) )
53	32, 52	eqtrd 2656	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` ( A mod ( 2 ^ N ) ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = (bits ` A ) )
54	53	oveq2d 6666	. . . 4 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd ( (bits ` ( A mod ( 2 ^ N ) ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) ) = ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` A ) ) )
55	30, 54	eqtrd 2656	. . 3 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` ( A mod ( 2 ^ N ) ) ) ) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` A ) ) )
56		sadid2 15191	. . . 4 |- ( ( (bits ` A ) i^i ( ZZ>= ` N ) ) C_ NN0 -> ( (/) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = ( (bits ` A ) i^i ( ZZ>= ` N ) ) )
57	28, 56	syl 17	. . 3 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (/) sadd ( (bits ` A ) i^i ( ZZ>= ` N ) ) ) = ( (bits ` A ) i^i ( ZZ>= ` N ) ) )
58	22, 55, 57	3eqtr3d 2664	. 2 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` -u ( A mod ( 2 ^ N ) ) ) sadd (bits ` A ) ) = ( (bits ` A ) i^i ( ZZ>= ` N ) ) )
59	1	zcnd 11483	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> A e. CC )
60	13, 59	addcomd 10238	. . . 4 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( -u ( A mod ( 2 ^ N ) ) + A ) = ( A + -u ( A mod ( 2 ^ N ) ) ) )
61	59, 14	negsubd 10398	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( A + -u ( A mod ( 2 ^ N ) ) ) = ( A - ( A mod ( 2 ^ N ) ) ) )
62	59, 14	subcld 10392	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( A - ( A mod ( 2 ^ N ) ) ) e. CC )
63	5	nncnd 11036	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( 2 ^ N ) e. CC )
64	5	nnne0d 11065	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( 2 ^ N ) =/= 0 )
65	62, 63, 64	divcan1d 10802	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( ( A - ( A mod ( 2 ^ N ) ) ) / ( 2 ^ N ) ) x. ( 2 ^ N ) ) = ( A - ( A mod ( 2 ^ N ) ) ) )
66	1	zred 11482	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN0 ) -> A e. RR )
67	5	nnrpd 11870	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( 2 ^ N ) e. RR+ )
68		moddiffl 12681	. . . . . . 7 |- ( ( A e. RR /\ ( 2 ^ N ) e. RR+ ) -> ( ( A - ( A mod ( 2 ^ N ) ) ) / ( 2 ^ N ) ) = ( |_ ` ( A / ( 2 ^ N ) ) ) )
69	66, 67, 68	syl2anc 693	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( A - ( A mod ( 2 ^ N ) ) ) / ( 2 ^ N ) ) = ( |_ ` ( A / ( 2 ^ N ) ) ) )
70	69	oveq1d 6665	. . . . 5 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( ( ( A - ( A mod ( 2 ^ N ) ) ) / ( 2 ^ N ) ) x. ( 2 ^ N ) ) = ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) )
71	61, 65, 70	3eqtr2d 2662	. . . 4 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( A + -u ( A mod ( 2 ^ N ) ) ) = ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) )
72	60, 71	eqtrd 2656	. . 3 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( -u ( A mod ( 2 ^ N ) ) + A ) = ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) )
73	72	fveq2d 6195	. 2 |- ( ( A e. ZZ /\ N e. NN0 ) -> (bits ` ( -u ( A mod ( 2 ^ N ) ) + A ) ) = (bits ` ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) ) )
74	10, 58, 73	3eqtr3d 2664	1 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( (bits ` A ) i^i ( ZZ>= ` N ) ) = (bits ` ( ( |_ ` ( A / ( 2 ^ N ) ) ) x. ( 2 ^ N ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832  ..^cfzo 12465   |_cfl 12591   mod cmo 12668   ^cexp 12860  bitscbits 15141   sadd csad 15142

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:  bitsuz  15196