Theorem 3dvds 12029

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 9355	. . 3 |- 3 e. ZZ
2	1	a1i 9	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 3 e. ZZ )
3		0zd 9338	. . . 4 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 0 e. ZZ )
4		nn0z 9346	. . . . 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 10523	. . 3 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> ( 0 ... N ) e. Fin )
7		ffvelcdm 5695	. . . . 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 9472	. . . . . 6 |- ; 1 0 e. NN
10	9	nnzi 9347	. . . . 5 |- ; 1 0 e. ZZ
11		elfznn0 10189	. . . . . 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 10646	. . . . 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 9454	. . 3 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( F ` k ) x. (; 1 0 ^ k ) ) e. ZZ )
16	6, 15	fsumzcl 11567	. 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 11567	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> sum_ k e. ( 0 ... N ) ( F ` k ) e. ZZ )
18	15, 8	zsubcld 9453	. . . 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 7972	. . . . . . . . . . . 12 |- 1 e. CC
20	9	nncni 9000	. . . . . . . . . . . 12 |- ; 1 0 e. CC
21	19, 20	negsubdi2i 8312	. . . . . . . . . . 11 |- -u ( 1 - ; 1 0 ) = (; 1 0 - 1 )
22		9p1e10 9459	. . . . . . . . . . . . 13 |- ( 9 + 1 ) = ; 1 0
23	22	eqcomi 2200	. . . . . . . . . . . 12 |- ; 1 0 = ( 9 + 1 )
24	23	oveq1i 5932	. . . . . . . . . . 11 |- (; 1 0 - 1 ) = ( ( 9 + 1 ) - 1 )
25		9cn 9078	. . . . . . . . . . . 12 |- 9 e. CC
26	25, 19	pncan3oi 8242	. . . . . . . . . . 11 |- ( ( 9 + 1 ) - 1 ) = 9
27	21, 24, 26	3eqtri 2221	. . . . . . . . . 10 |- -u ( 1 - ; 1 0 ) = 9
28		3t3e9 9148	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
29	27, 28	eqtr4i 2220	. . . . . . . . 9 |- -u ( 1 - ; 1 0 ) = ( 3 x. 3 )
30	20	a1i 9	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> ; 1 0 e. CC )
31		1re 8025	. . . . . . . . . . . . . . . . 17 |- 1 e. RR
32		10re 9475	. . . . . . . . . . . . . . . . 17 |- ; 1 0 e. RR
33		1lt10 9595	. . . . . . . . . . . . . . . . 17 |- 1 < ; 1 0
34	31, 32, 33	gtapii 8661	. . . . . . . . . . . . . . . 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 11672	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> sum_ j e. ( 0 ... ( k - 1 ) ) (; 1 0 ^ j ) = ( ( 1 - (; 1 0 ^ k ) ) / ( 1 - ; 1 0 ) ) )
38		0zd 9338	. . . . . . . . . . . . . . . 16 |- ( k e. NN0 -> 0 e. ZZ )
39		nn0z 9346	. . . . . . . . . . . . . . . . 17 |- ( k e. NN0 -> k e. ZZ )
40		peano2zm 9364	. . . . . . . . . . . . . . . . 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 10523	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> ( 0 ... ( k - 1 ) ) e. Fin )
43		elfznn0 10189	. . . . . . . . . . . . . . . . 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 10646	. . . . . . . . . . . . . . . 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 11567	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> sum_ j e. ( 0 ... ( k - 1 ) ) (; 1 0 ^ j ) e. ZZ )
48	37, 47	eqeltrrd 2274	. . . . . . . . . . . . 13 |- ( k e. NN0 -> ( ( 1 - (; 1 0 ^ k ) ) / ( 1 - ; 1 0 ) ) e. ZZ )
49		1z 9352	. . . . . . . . . . . . . . 15 |- 1 e. ZZ
50		zsubcl 9367	. . . . . . . . . . . . . . 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 8123	. . . . . . . . . . . . . . 15 |- 1 =/= ; 1 0
53	19, 20	subeq0i 8306	. . . . . . . . . . . . . . . 16 |- ( ( 1 - ; 1 0 ) = 0 <-> 1 = ; 1 0 )
54	53	necon3bii 2405	. . . . . . . . . . . . . . 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 9367	. . . . . . . . . . . . . . 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 11955	. . . . . . . . . . . . . 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 1352	. . . . . . . . . . . . 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 9449	. . . . . . . . . . . . 13 |- ( k e. NN0 -> (; 1 0 ^ k ) e. CC )
63		negsubdi2 8285	. . . . . . . . . . . . 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 4061	. . . . . . . . . . 11 |- ( k e. NN0 -> ( 1 - ; 1 0 ) || -u ( (; 1 0 ^ k ) - 1 ) )
66		peano2zm 9364	. . . . . . . . . . . . 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 11973	. . . . . . . . . . . 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 11972	. . . . . . . . . . 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 4069	. . . . . . . 8 |- ( k e. NN0 -> ( 3 x. 3 ) || ( (; 1 0 ^ k ) - 1 ) )
75		muldvds1 11981	. . . . . . . . 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 1352	. . . . . . . 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 11998	. . . . . . 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 1353	. . . . . 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 9449	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( F ` k ) e. CC )
84	14	zcnd 9449	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> (; 1 0 ^ k ) e. CC )
85	83, 84	muls1d 8444	. . . . 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 4059	. . . 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 12007	. . 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 9449	. . . 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 11617	. . 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 4059	. 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 12000	. 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 1252	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 1364   e. wcel 2167   =/= wne 2367   class class class wbr 4033   -->wf 5254   ` cfv 5258  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   - cmin 8197   -ucneg 8198   # cap 8608   / cdiv 8699   3c3 9042   9c9 9048   NN0cn0 9249   ZZcz 9326  ;cdc 9457   ...cfz 10083   ^cexp 10630   sum_csu 11518   || cdvds 11952

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-n0 9250  df-z 9327  df-dec 9458  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519  df-dvds 11953

This theorem is referenced by: (None)