Theorem smumul 15215

Description: For sequences that correspond to valid integers, the sequence multiplication function produces the sequence for the product. This is effectively a proof of the correctness of the multiplication process, implemented in terms of logic gates for df-sad 15173, whose correctness is verified in sadadd 15189.

Outside this range, the sequences cannot be representing integers, but the smul function still "works". This extended function is best interpreted in terms of the ring structure of the 2-adic integers. (Contributed by Mario Carneiro, 22-Sep-2016.)

Assertion

Ref	Expression
smumul	|- ( ( A e. ZZ /\ B e. ZZ ) -> ( (bits ` A ) smul (bits ` B ) ) = (bits ` ( A x. B ) ) )

Proof of Theorem smumul

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bitsss 15148	. . . . . 6 |- (bits ` A ) C_ NN0
2		bitsss 15148	. . . . . 6 |- (bits ` B ) C_ NN0
3		smucl 15206	. . . . . 6 |- ( ( (bits ` A ) C_ NN0 /\ (bits ` B ) C_ NN0 ) -> ( (bits ` A ) smul (bits ` B ) ) C_ NN0 )
4	1, 2, 3	mp2an 708	. . . . 5 |- ( (bits ` A ) smul (bits ` B ) ) C_ NN0
5	4	sseli 3599	. . . 4 |- ( k e. ( (bits ` A ) smul (bits ` B ) ) -> k e. NN0 )
6	5	a1i 11	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( k e. ( (bits ` A ) smul (bits ` B ) ) -> k e. NN0 ) )
7		bitsss 15148	. . . . 5 |- (bits ` ( A x. B ) ) C_ NN0
8	7	sseli 3599	. . . 4 |- ( k e. (bits ` ( A x. B ) ) -> k e. NN0 )
9	8	a1i 11	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( k e. (bits ` ( A x. B ) ) -> k e. NN0 ) )
10		simpll 790	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> A e. ZZ )
11		simplr 792	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> B e. ZZ )
12		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> k e. NN0 )
13		1nn0 11308	. . . . . . . . . . . . . 14 |- 1 e. NN0
14	13	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> 1 e. NN0 )
15	12, 14	nn0addcld 11355	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( k + 1 ) e. NN0 )
16	10, 11, 15	smumullem 15214	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) smul (bits ` B ) ) = (bits ` ( ( A mod ( 2 ^ ( k + 1 ) ) ) x. B ) ) )
17	16	ineq1d 3813	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) smul (bits ` B ) ) i^i ( 0..^ ( k + 1 ) ) ) = ( (bits ` ( ( A mod ( 2 ^ ( k + 1 ) ) ) x. B ) ) i^i ( 0..^ ( k + 1 ) ) ) )
18		2nn 11185	. . . . . . . . . . . . . . . 16 |- 2 e. NN
19	18	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> 2 e. NN )
20	19, 15	nnexpcld 13030	. . . . . . . . . . . . . 14 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( 2 ^ ( k + 1 ) ) e. NN )
21	10, 20	zmodcld 12691	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( A mod ( 2 ^ ( k + 1 ) ) ) e. NN0 )
22	21	nn0zd 11480	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( A mod ( 2 ^ ( k + 1 ) ) ) e. ZZ )
23	22, 11	zmulcld 11488	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( A mod ( 2 ^ ( k + 1 ) ) ) x. B ) e. ZZ )
24		bitsmod 15158	. . . . . . . . . . 11 |- ( ( ( ( A mod ( 2 ^ ( k + 1 ) ) ) x. B ) e. ZZ /\ ( k + 1 ) e. NN0 ) -> (bits ` ( ( ( A mod ( 2 ^ ( k + 1 ) ) ) x. B ) mod ( 2 ^ ( k + 1 ) ) ) ) = ( (bits ` ( ( A mod ( 2 ^ ( k + 1 ) ) ) x. B ) ) i^i ( 0..^ ( k + 1 ) ) ) )
25	23, 15, 24	syl2anc 693	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> (bits ` ( ( ( A mod ( 2 ^ ( k + 1 ) ) ) x. B ) mod ( 2 ^ ( k + 1 ) ) ) ) = ( (bits ` ( ( A mod ( 2 ^ ( k + 1 ) ) ) x. B ) ) i^i ( 0..^ ( k + 1 ) ) ) )
26	17, 25	eqtr4d 2659	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) smul (bits ` B ) ) i^i ( 0..^ ( k + 1 ) ) ) = (bits ` ( ( ( A mod ( 2 ^ ( k + 1 ) ) ) x. B ) mod ( 2 ^ ( k + 1 ) ) ) ) )
27		inass 3823	. . . . . . . . . . . . 13 |- ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) i^i ( 0..^ ( k + 1 ) ) ) = ( (bits ` A ) i^i ( ( 0..^ ( k + 1 ) ) i^i ( 0..^ ( k + 1 ) ) ) )
28		inidm 3822	. . . . . . . . . . . . . 14 |- ( ( 0..^ ( k + 1 ) ) i^i ( 0..^ ( k + 1 ) ) ) = ( 0..^ ( k + 1 ) )
29	28	ineq2i 3811	. . . . . . . . . . . . 13 |- ( (bits ` A ) i^i ( ( 0..^ ( k + 1 ) ) i^i ( 0..^ ( k + 1 ) ) ) ) = ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) )
30	27, 29	eqtri 2644	. . . . . . . . . . . 12 |- ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) i^i ( 0..^ ( k + 1 ) ) ) = ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) )
31	30	oveq1i 6660	. . . . . . . . . . 11 |- ( ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) i^i ( 0..^ ( k + 1 ) ) ) smul ( (bits ` B ) i^i ( 0..^ ( k + 1 ) ) ) ) = ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) smul ( (bits ` B ) i^i ( 0..^ ( k + 1 ) ) ) )
32	31	ineq1i 3810	. . . . . . . . . 10 |- ( ( ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) i^i ( 0..^ ( k + 1 ) ) ) smul ( (bits ` B ) i^i ( 0..^ ( k + 1 ) ) ) ) i^i ( 0..^ ( k + 1 ) ) ) = ( ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) smul ( (bits ` B ) i^i ( 0..^ ( k + 1 ) ) ) ) i^i ( 0..^ ( k + 1 ) ) )
33		inss1 3833	. . . . . . . . . . . 12 |- ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) C_ (bits ` A )
34	1	a1i 11	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> (bits ` A ) C_ NN0 )
35	33, 34	syl5ss 3614	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) C_ NN0 )
36	2	a1i 11	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> (bits ` B ) C_ NN0 )
37	35, 36, 15	smueq 15213	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) smul (bits ` B ) ) i^i ( 0..^ ( k + 1 ) ) ) = ( ( ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) i^i ( 0..^ ( k + 1 ) ) ) smul ( (bits ` B ) i^i ( 0..^ ( k + 1 ) ) ) ) i^i ( 0..^ ( k + 1 ) ) ) )
38	34, 36, 15	smueq 15213	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( (bits ` A ) smul (bits ` B ) ) i^i ( 0..^ ( k + 1 ) ) ) = ( ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) smul ( (bits ` B ) i^i ( 0..^ ( k + 1 ) ) ) ) i^i ( 0..^ ( k + 1 ) ) ) )
39	32, 37, 38	3eqtr4a 2682	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( ( (bits ` A ) i^i ( 0..^ ( k + 1 ) ) ) smul (bits ` B ) ) i^i ( 0..^ ( k + 1 ) ) ) = ( ( (bits ` A ) smul (bits ` B ) ) i^i ( 0..^ ( k + 1 ) ) ) )
40	20	nnrpd 11870	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( 2 ^ ( k + 1 ) ) e. RR+ )
41	10	zred 11482	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> A e. RR )
42		modabs2 12704	. . . . . . . . . . . 12 |- ( ( A e. RR /\ ( 2 ^ ( k + 1 ) ) e. RR+ ) -> ( ( A mod ( 2 ^ ( k + 1 ) ) ) mod ( 2 ^ ( k + 1 ) ) ) = ( A mod ( 2 ^ ( k + 1 ) ) ) )
43	41, 40, 42	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( A mod ( 2 ^ ( k + 1 ) ) ) mod ( 2 ^ ( k + 1 ) ) ) = ( A mod ( 2 ^ ( k + 1 ) ) ) )
44		eqidd 2623	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( B mod ( 2 ^ ( k + 1 ) ) ) = ( B mod ( 2 ^ ( k + 1 ) ) ) )
45	22, 10, 11, 11, 40, 43, 44	modmul12d 12724	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( ( A mod ( 2 ^ ( k + 1 ) ) ) x. B ) mod ( 2 ^ ( k + 1 ) ) ) = ( ( A x. B ) mod ( 2 ^ ( k + 1 ) ) ) )
46	45	fveq2d 6195	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> (bits ` ( ( ( A mod ( 2 ^ ( k + 1 ) ) ) x. B ) mod ( 2 ^ ( k + 1 ) ) ) ) = (bits ` ( ( A x. B ) mod ( 2 ^ ( k + 1 ) ) ) ) )
47	26, 39, 46	3eqtr3d 2664	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( (bits ` A ) smul (bits ` B ) ) i^i ( 0..^ ( k + 1 ) ) ) = (bits ` ( ( A x. B ) mod ( 2 ^ ( k + 1 ) ) ) ) )
48	10, 11	zmulcld 11488	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( A x. B ) e. ZZ )
49		bitsmod 15158	. . . . . . . . 9 |- ( ( ( A x. B ) e. ZZ /\ ( k + 1 ) e. NN0 ) -> (bits ` ( ( A x. B ) mod ( 2 ^ ( k + 1 ) ) ) ) = ( (bits ` ( A x. B ) ) i^i ( 0..^ ( k + 1 ) ) ) )
50	48, 15, 49	syl2anc 693	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> (bits ` ( ( A x. B ) mod ( 2 ^ ( k + 1 ) ) ) ) = ( (bits ` ( A x. B ) ) i^i ( 0..^ ( k + 1 ) ) ) )
51	47, 50	eqtrd 2656	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( (bits ` A ) smul (bits ` B ) ) i^i ( 0..^ ( k + 1 ) ) ) = ( (bits ` ( A x. B ) ) i^i ( 0..^ ( k + 1 ) ) ) )
52	51	eleq2d 2687	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( k e. ( ( (bits ` A ) smul (bits ` B ) ) i^i ( 0..^ ( k + 1 ) ) ) <-> k e. ( (bits ` ( A x. B ) ) i^i ( 0..^ ( k + 1 ) ) ) ) )
53		elin 3796	. . . . . 6 |- ( k e. ( ( (bits ` A ) smul (bits ` B ) ) i^i ( 0..^ ( k + 1 ) ) ) <-> ( k e. ( (bits ` A ) smul (bits ` B ) ) /\ k e. ( 0..^ ( k + 1 ) ) ) )
54		elin 3796	. . . . . 6 |- ( k e. ( (bits ` ( A x. B ) ) i^i ( 0..^ ( k + 1 ) ) ) <-> ( k e. (bits ` ( A x. B ) ) /\ k e. ( 0..^ ( k + 1 ) ) ) )
55	52, 53, 54	3bitr3g 302	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( ( k e. ( (bits ` A ) smul (bits ` B ) ) /\ k e. ( 0..^ ( k + 1 ) ) ) <-> ( k e. (bits ` ( A x. B ) ) /\ k e. ( 0..^ ( k + 1 ) ) ) ) )
56		nn0uz 11722	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
57	12, 56	syl6eleq 2711	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> k e. ( ZZ>= ` 0 ) )
58		eluzfz2b 12350	. . . . . . . 8 |- ( k e. ( ZZ>= ` 0 ) <-> k e. ( 0 ... k ) )
59	57, 58	sylib 208	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> k e. ( 0 ... k ) )
60	12	nn0zd 11480	. . . . . . . 8 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> k e. ZZ )
61		fzval3 12536	. . . . . . . 8 |- ( k e. ZZ -> ( 0 ... k ) = ( 0..^ ( k + 1 ) ) )
62	60, 61	syl 17	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( 0 ... k ) = ( 0..^ ( k + 1 ) ) )
63	59, 62	eleqtrd 2703	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> k e. ( 0..^ ( k + 1 ) ) )
64	63	biantrud 528	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( k e. ( (bits ` A ) smul (bits ` B ) ) <-> ( k e. ( (bits ` A ) smul (bits ` B ) ) /\ k e. ( 0..^ ( k + 1 ) ) ) ) )
65	63	biantrud 528	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( k e. (bits ` ( A x. B ) ) <-> ( k e. (bits ` ( A x. B ) ) /\ k e. ( 0..^ ( k + 1 ) ) ) ) )
66	55, 64, 65	3bitr4d 300	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ k e. NN0 ) -> ( k e. ( (bits ` A ) smul (bits ` B ) ) <-> k e. (bits ` ( A x. B ) ) ) )
67	66	ex 450	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( k e. NN0 -> ( k e. ( (bits ` A ) smul (bits ` B ) ) <-> k e. (bits ` ( A x. B ) ) ) ) )
68	6, 9, 67	pm5.21ndd 369	. 2 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( k e. ( (bits ` A ) smul (bits ` B ) ) <-> k e. (bits ` ( A x. B ) ) ) )
69	68	eqrdv 2620	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( (bits ` A ) smul (bits ` B ) ) = (bits ` ( A x. B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326  ..^cfzo 12465   mod cmo 12668   ^cexp 12860  bitscbits 15141   smul csmu 15143

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  df-smu 15198

This theorem is referenced by: (None)