Theorem 3dvds 12341

Description: A rule for divisibility by 3 of a number written in base 10. This is Metamath 100 proof #85. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.) (Revised by AV, 8-Sep-2021.)

Assertion

Ref	Expression
3dvds	|- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> ( 3 || sum_ k e. ( 0 ... N ) ( ( F ` k ) x. (; 1 0 ^ k ) ) <-> 3 || sum_ k e. ( 0 ... N ) ( F ` k ) ) )

Distinct variable groups:   k, F   k, N

Proof of Theorem 3dvds

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3z 9443	. . 3 |- 3 e. ZZ
2	1	a1i 9	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 3 e. ZZ )
3		0zd 9426	. . . 4 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 0 e. ZZ )
4		nn0z 9434	. . . . 5 |- ( N e. NN0 -> N e. ZZ )
5	4	adantr 276	. . . 4 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> N e. ZZ )
6	3, 5	fzfigd 10620	. . 3 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> ( 0 ... N ) e. Fin )
7		ffvelcdm 5741	. . . . 5 |- ( ( F : ( 0 ... N ) --> ZZ /\ k e. ( 0 ... N ) ) -> ( F ` k ) e. ZZ )
8	7	adantll 476	. . . 4 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( F ` k ) e. ZZ )
9		10nn 9561	. . . . . 6 |- ; 1 0 e. NN
10	9	nnzi 9435	. . . . 5 |- ; 1 0 e. ZZ
11		elfznn0 10278	. . . . . 6 |- ( k e. ( 0 ... N ) -> k e. NN0 )
12	11	adantl 277	. . . . 5 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> k e. NN0 )
13		zexpcl 10743	. . . . 5 |- ( (; 1 0 e. ZZ /\ k e. NN0 ) -> (; 1 0 ^ k ) e. ZZ )
14	10, 12, 13	sylancr 414	. . . 4 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> (; 1 0 ^ k ) e. ZZ )
15	8, 14	zmulcld 9543	. . 3 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( F ` k ) x. (; 1 0 ^ k ) ) e. ZZ )
16	6, 15	fsumzcl 11879	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> sum_ k e. ( 0 ... N ) ( ( F ` k ) x. (; 1 0 ^ k ) ) e. ZZ )
17	6, 8	fsumzcl 11879	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> sum_ k e. ( 0 ... N ) ( F ` k ) e. ZZ )
18	15, 8	zsubcld 9542	. . . 4 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( ( F ` k ) x. (; 1 0 ^ k ) ) - ( F ` k ) ) e. ZZ )
19		ax-1cn 8060	. . . . . . . . . . . 12 |- 1 e. CC
20	9	nncni 9088	. . . . . . . . . . . 12 |- ; 1 0 e. CC
21	19, 20	negsubdi2i 8400	. . . . . . . . . . 11 |- -u ( 1 - ; 1 0 ) = (; 1 0 - 1 )
22		9p1e10 9548	. . . . . . . . . . . . 13 |- ( 9 + 1 ) = ; 1 0
23	22	eqcomi 2213	. . . . . . . . . . . 12 |- ; 1 0 = ( 9 + 1 )
24	23	oveq1i 5984	. . . . . . . . . . 11 |- (; 1 0 - 1 ) = ( ( 9 + 1 ) - 1 )
25		9cn 9166	. . . . . . . . . . . 12 |- 9 e. CC
26	25, 19	pncan3oi 8330	. . . . . . . . . . 11 |- ( ( 9 + 1 ) - 1 ) = 9
27	21, 24, 26	3eqtri 2234	. . . . . . . . . 10 |- -u ( 1 - ; 1 0 ) = 9
28		3t3e9 9236	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
29	27, 28	eqtr4i 2233	. . . . . . . . 9 |- -u ( 1 - ; 1 0 ) = ( 3 x. 3 )
30	20	a1i 9	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> ; 1 0 e. CC )
31		1re 8113	. . . . . . . . . . . . . . . . 17 |- 1 e. RR
32		10re 9564	. . . . . . . . . . . . . . . . 17 |- ; 1 0 e. RR
33		1lt10 9684	. . . . . . . . . . . . . . . . 17 |- 1 < ; 1 0
34	31, 32, 33	gtapii 8749	. . . . . . . . . . . . . . . 16 |- ; 1 0 # 1
35	34	a1i 9	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> ; 1 0 # 1 )
36		id 19	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> k e. NN0 )
37	30, 35, 36	geoserap 11984	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> sum_ j e. ( 0 ... ( k - 1 ) ) (; 1 0 ^ j ) = ( ( 1 - (; 1 0 ^ k ) ) / ( 1 - ; 1 0 ) ) )
38		0zd 9426	. . . . . . . . . . . . . . . 16 |- ( k e. NN0 -> 0 e. ZZ )
39		nn0z 9434	. . . . . . . . . . . . . . . . 17 |- ( k e. NN0 -> k e. ZZ )
40		peano2zm 9452	. . . . . . . . . . . . . . . . 17 |- ( k e. ZZ -> ( k - 1 ) e. ZZ )
41	39, 40	syl 14	. . . . . . . . . . . . . . . 16 |- ( k e. NN0 -> ( k - 1 ) e. ZZ )
42	38, 41	fzfigd 10620	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> ( 0 ... ( k - 1 ) ) e. Fin )
43		elfznn0 10278	. . . . . . . . . . . . . . . . 17 |- ( j e. ( 0 ... ( k - 1 ) ) -> j e. NN0 )
44	43	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN0 /\ j e. ( 0 ... ( k - 1 ) ) ) -> j e. NN0 )
45		zexpcl 10743	. . . . . . . . . . . . . . . 16 |- ( (; 1 0 e. ZZ /\ j e. NN0 ) -> (; 1 0 ^ j ) e. ZZ )
46	10, 44, 45	sylancr 414	. . . . . . . . . . . . . . 15 |- ( ( k e. NN0 /\ j e. ( 0 ... ( k - 1 ) ) ) -> (; 1 0 ^ j ) e. ZZ )
47	42, 46	fsumzcl 11879	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> sum_ j e. ( 0 ... ( k - 1 ) ) (; 1 0 ^ j ) e. ZZ )
48	37, 47	eqeltrrd 2287	. . . . . . . . . . . . 13 |- ( k e. NN0 -> ( ( 1 - (; 1 0 ^ k ) ) / ( 1 - ; 1 0 ) ) e. ZZ )
49		1z 9440	. . . . . . . . . . . . . . 15 |- 1 e. ZZ
50		zsubcl 9455	. . . . . . . . . . . . . . 15 |- ( ( 1 e. ZZ /\ ; 1 0 e. ZZ ) -> ( 1 - ; 1 0 ) e. ZZ )
51	49, 10, 50	mp2an 426	. . . . . . . . . . . . . 14 |- ( 1 - ; 1 0 ) e. ZZ
52	31, 33	ltneii 8211	. . . . . . . . . . . . . . 15 |- 1 =/= ; 1 0
53	19, 20	subeq0i 8394	. . . . . . . . . . . . . . . 16 |- ( ( 1 - ; 1 0 ) = 0 <-> 1 = ; 1 0 )
54	53	necon3bii 2418	. . . . . . . . . . . . . . 15 |- ( ( 1 - ; 1 0 ) =/= 0 <-> 1 =/= ; 1 0 )
55	52, 54	mpbir 146	. . . . . . . . . . . . . 14 |- ( 1 - ; 1 0 ) =/= 0
56	10, 36, 13	sylancr 414	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> (; 1 0 ^ k ) e. ZZ )
57		zsubcl 9455	. . . . . . . . . . . . . . 15 |- ( ( 1 e. ZZ /\ (; 1 0 ^ k ) e. ZZ ) -> ( 1 - (; 1 0 ^ k ) ) e. ZZ )
58	49, 56, 57	sylancr 414	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> ( 1 - (; 1 0 ^ k ) ) e. ZZ )
59		dvdsval2 12267	. . . . . . . . . . . . . 14 |- ( ( ( 1 - ; 1 0 ) e. ZZ /\ ( 1 - ; 1 0 ) =/= 0 /\ ( 1 - (; 1 0 ^ k ) ) e. ZZ ) -> ( ( 1 - ; 1 0 ) || ( 1 - (; 1 0 ^ k ) ) <-> ( ( 1 - (; 1 0 ^ k ) ) / ( 1 - ; 1 0 ) ) e. ZZ ) )
60	51, 55, 58, 59	mp3an12i 1356	. . . . . . . . . . . . 13 |- ( k e. NN0 -> ( ( 1 - ; 1 0 ) || ( 1 - (; 1 0 ^ k ) ) <-> ( ( 1 - (; 1 0 ^ k ) ) / ( 1 - ; 1 0 ) ) e. ZZ ) )
61	48, 60	mpbird 167	. . . . . . . . . . . 12 |- ( k e. NN0 -> ( 1 - ; 1 0 ) || ( 1 - (; 1 0 ^ k ) ) )
62	56	zcnd 9538	. . . . . . . . . . . . 13 |- ( k e. NN0 -> (; 1 0 ^ k ) e. CC )
63		negsubdi2 8373	. . . . . . . . . . . . 13 |- ( ( (; 1 0 ^ k ) e. CC /\ 1 e. CC ) -> -u ( (; 1 0 ^ k ) - 1 ) = ( 1 - (; 1 0 ^ k ) ) )
64	62, 19, 63	sylancl 413	. . . . . . . . . . . 12 |- ( k e. NN0 -> -u ( (; 1 0 ^ k ) - 1 ) = ( 1 - (; 1 0 ^ k ) ) )
65	61, 64	breqtrrd 4090	. . . . . . . . . . 11 |- ( k e. NN0 -> ( 1 - ; 1 0 ) || -u ( (; 1 0 ^ k ) - 1 ) )
66		peano2zm 9452	. . . . . . . . . . . . 13 |- ( (; 1 0 ^ k ) e. ZZ -> ( (; 1 0 ^ k ) - 1 ) e. ZZ )
67	56, 66	syl 14	. . . . . . . . . . . 12 |- ( k e. NN0 -> ( (; 1 0 ^ k ) - 1 ) e. ZZ )
68		dvdsnegb 12285	. . . . . . . . . . . 12 |- ( ( ( 1 - ; 1 0 ) e. ZZ /\ ( (; 1 0 ^ k ) - 1 ) e. ZZ ) -> ( ( 1 - ; 1 0 ) || ( (; 1 0 ^ k ) - 1 ) <-> ( 1 - ; 1 0 ) || -u ( (; 1 0 ^ k ) - 1 ) ) )
69	51, 67, 68	sylancr 414	. . . . . . . . . . 11 |- ( k e. NN0 -> ( ( 1 - ; 1 0 ) || ( (; 1 0 ^ k ) - 1 ) <-> ( 1 - ; 1 0 ) || -u ( (; 1 0 ^ k ) - 1 ) ) )
70	65, 69	mpbird 167	. . . . . . . . . 10 |- ( k e. NN0 -> ( 1 - ; 1 0 ) || ( (; 1 0 ^ k ) - 1 ) )
71		negdvdsb 12284	. . . . . . . . . . 11 |- ( ( ( 1 - ; 1 0 ) e. ZZ /\ ( (; 1 0 ^ k ) - 1 ) e. ZZ ) -> ( ( 1 - ; 1 0 ) || ( (; 1 0 ^ k ) - 1 ) <-> -u ( 1 - ; 1 0 ) || ( (; 1 0 ^ k ) - 1 ) ) )
72	51, 67, 71	sylancr 414	. . . . . . . . . 10 |- ( k e. NN0 -> ( ( 1 - ; 1 0 ) || ( (; 1 0 ^ k ) - 1 ) <-> -u ( 1 - ; 1 0 ) || ( (; 1 0 ^ k ) - 1 ) ) )
73	70, 72	mpbid 147	. . . . . . . . 9 |- ( k e. NN0 -> -u ( 1 - ; 1 0 ) || ( (; 1 0 ^ k ) - 1 ) )
74	29, 73	eqbrtrrid 4098	. . . . . . . 8 |- ( k e. NN0 -> ( 3 x. 3 ) || ( (; 1 0 ^ k ) - 1 ) )
75		muldvds1 12293	. . . . . . . . 9 |- ( ( 3 e. ZZ /\ 3 e. ZZ /\ ( (; 1 0 ^ k ) - 1 ) e. ZZ ) -> ( ( 3 x. 3 ) || ( (; 1 0 ^ k ) - 1 ) -> 3 || ( (; 1 0 ^ k ) - 1 ) ) )
76	1, 1, 67, 75	mp3an12i 1356	. . . . . . . 8 |- ( k e. NN0 -> ( ( 3 x. 3 ) || ( (; 1 0 ^ k ) - 1 ) -> 3 || ( (; 1 0 ^ k ) - 1 ) ) )
77	74, 76	mpd 13	. . . . . . 7 |- ( k e. NN0 -> 3 || ( (; 1 0 ^ k ) - 1 ) )
78	12, 77	syl 14	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> 3 || ( (; 1 0 ^ k ) - 1 ) )
79	14, 66	syl 14	. . . . . . 7 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( (; 1 0 ^ k ) - 1 ) e. ZZ )
80		dvdsmultr2 12310	. . . . . . 7 |- ( ( 3 e. ZZ /\ ( F ` k ) e. ZZ /\ ( (; 1 0 ^ k ) - 1 ) e. ZZ ) -> ( 3 || ( (; 1 0 ^ k ) - 1 ) -> 3 || ( ( F ` k ) x. ( (; 1 0 ^ k ) - 1 ) ) ) )
81	1, 8, 79, 80	mp3an2i 1357	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( 3 || ( (; 1 0 ^ k ) - 1 ) -> 3 || ( ( F ` k ) x. ( (; 1 0 ^ k ) - 1 ) ) ) )
82	78, 81	mpd 13	. . . . 5 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> 3 || ( ( F ` k ) x. ( (; 1 0 ^ k ) - 1 ) ) )
83	8	zcnd 9538	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( F ` k ) e. CC )
84	14	zcnd 9538	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> (; 1 0 ^ k ) e. CC )
85	83, 84	muls1d 8532	. . . . 5 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( F ` k ) x. ( (; 1 0 ^ k ) - 1 ) ) = ( ( ( F ` k ) x. (; 1 0 ^ k ) ) - ( F ` k ) ) )
86	82, 85	breqtrd 4088	. . . 4 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> 3 || ( ( ( F ` k ) x. (; 1 0 ^ k ) ) - ( F ` k ) ) )
87	6, 2, 18, 86	fsumdvds 12319	. . 3 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 3 || sum_ k e. ( 0 ... N ) ( ( ( F ` k ) x. (; 1 0 ^ k ) ) - ( F ` k ) ) )
88	15	zcnd 9538	. . . 4 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( F ` k ) x. (; 1 0 ^ k ) ) e. CC )
89	6, 88, 83	fsumsub 11929	. . 3 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> sum_ k e. ( 0 ... N ) ( ( ( F ` k ) x. (; 1 0 ^ k ) ) - ( F ` k ) ) = ( sum_ k e. ( 0 ... N ) ( ( F ` k ) x. (; 1 0 ^ k ) ) - sum_ k e. ( 0 ... N ) ( F ` k ) ) )
90	87, 89	breqtrd 4088	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 3 || ( sum_ k e. ( 0 ... N ) ( ( F ` k ) x. (; 1 0 ^ k ) ) - sum_ k e. ( 0 ... N ) ( F ` k ) ) )
91		dvdssub2 12312	. 2 |- ( ( ( 3 e. ZZ /\ sum_ k e. ( 0 ... N ) ( ( F ` k ) x. (; 1 0 ^ k ) ) e. ZZ /\ sum_ k e. ( 0 ... N ) ( F ` k ) e. ZZ ) /\ 3 || ( sum_ k e. ( 0 ... N ) ( ( F ` k ) x. (; 1 0 ^ k ) ) - sum_ k e. ( 0 ... N ) ( F ` k ) ) ) -> ( 3 || sum_ k e. ( 0 ... N ) ( ( F ` k ) x. (; 1 0 ^ k ) ) <-> 3 || sum_ k e. ( 0 ... N ) ( F ` k ) ) )
92	2, 16, 17, 90, 91	syl31anc 1255	1 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> ( 3 || sum_ k e. ( 0 ... N ) ( ( F ` k ) x. (; 1 0 ^ k ) ) <-> 3 || sum_ k e. ( 0 ... N ) ( F ` k ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1375   e. wcel 2180   =/= wne 2380   class class class wbr 4062   -->wf 5290   ` cfv 5294  (class class class)co 5974   CCcc 7965   0cc0 7967   1c1 7968   + caddc 7970   x. cmul 7972   - cmin 8285   -ucneg 8286   # cap 8696   / cdiv 8787   3c3 9130   9c9 9136   NN0cn0 9337   ZZcz 9414  ;cdc 9546   ...cfz 10172   ^cexp 10727   sum_csu 11830   || cdvds 12264

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  ax-caucvg 8087

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-isom 5303  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-frec 6507  df-1o 6532  df-oadd 6536  df-er 6650  df-en 6858  df-dom 6859  df-fin 6860  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-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-z 9415  df-dec 9547  df-uz 9691  df-q 9783  df-rp 9818  df-fz 10173  df-fzo 10307  df-seqfrec 10637  df-exp 10728  df-ihash 10965  df-cj 11319  df-re 11320  df-im 11321  df-rsqrt 11475  df-abs 11476  df-clim 11756  df-sumdc 11831  df-dvds 12265

This theorem is referenced by: (None)