Theorem 3dvds 12578

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 9626	. . 3 |- 3 e. ZZ
2	1	a1i 9	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 3 e. ZZ )
3		0zd 9609	. . . 4 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 0 e. ZZ )
4		nn0z 9617	. . . . 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 10820	. . 3 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> ( 0 ... N ) e. Fin )
7		ffvelcdm 5815	. . . . 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 9745	. . . . . 6 |- ; 1 0 e. NN
10	9	nnzi 9618	. . . . 5 |- ; 1 0 e. ZZ
11		elfznn0 10473	. . . . . 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 10943	. . . . 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 9727	. . 3 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( F ` k ) x. (; 1 0 ^ k ) ) e. ZZ )
16	6, 15	fsumzcl 12116	. 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 12116	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> sum_ k e. ( 0 ... N ) ( F ` k ) e. ZZ )
18	15, 8	zsubcld 9726	. . . 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 8236	. . . . . . . . . . . 12 |- 1 e. CC
20	9	nncni 9267	. . . . . . . . . . . 12 |- ; 1 0 e. CC
21	19, 20	negsubdi2i 8576	. . . . . . . . . . 11 |- -u ( 1 - ; 1 0 ) = (; 1 0 - 1 )
22		9p1e10 9732	. . . . . . . . . . . . 13 |- ( 9 + 1 ) = ; 1 0
23	22	eqcomi 2238	. . . . . . . . . . . 12 |- ; 1 0 = ( 9 + 1 )
24	23	oveq1i 6068	. . . . . . . . . . 11 |- (; 1 0 - 1 ) = ( ( 9 + 1 ) - 1 )
25		9cn 9345	. . . . . . . . . . . 12 |- 9 e. CC
26	25, 19	pncan3oi 8506	. . . . . . . . . . 11 |- ( ( 9 + 1 ) - 1 ) = 9
27	21, 24, 26	3eqtri 2259	. . . . . . . . . 10 |- -u ( 1 - ; 1 0 ) = 9
28		3t3e9 9415	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
29	27, 28	eqtr4i 2258	. . . . . . . . 9 |- -u ( 1 - ; 1 0 ) = ( 3 x. 3 )
30	20	a1i 9	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> ; 1 0 e. CC )
31		1re 8289	. . . . . . . . . . . . . . . . 17 |- 1 e. RR
32		10re 9748	. . . . . . . . . . . . . . . . 17 |- ; 1 0 e. RR
33		1lt10 9868	. . . . . . . . . . . . . . . . 17 |- 1 < ; 1 0
34	31, 32, 33	gtapii 8926	. . . . . . . . . . . . . . . 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 12221	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> sum_ j e. ( 0 ... ( k - 1 ) ) (; 1 0 ^ j ) = ( ( 1 - (; 1 0 ^ k ) ) / ( 1 - ; 1 0 ) ) )
38		0zd 9609	. . . . . . . . . . . . . . . 16 |- ( k e. NN0 -> 0 e. ZZ )
39		nn0z 9617	. . . . . . . . . . . . . . . . 17 |- ( k e. NN0 -> k e. ZZ )
40		peano2zm 9635	. . . . . . . . . . . . . . . . 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 10820	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> ( 0 ... ( k - 1 ) ) e. Fin )
43		elfznn0 10473	. . . . . . . . . . . . . . . . 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 10943	. . . . . . . . . . . . . . . 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 12116	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> sum_ j e. ( 0 ... ( k - 1 ) ) (; 1 0 ^ j ) e. ZZ )
48	37, 47	eqeltrrd 2312	. . . . . . . . . . . . 13 |- ( k e. NN0 -> ( ( 1 - (; 1 0 ^ k ) ) / ( 1 - ; 1 0 ) ) e. ZZ )
49		1z 9623	. . . . . . . . . . . . . . 15 |- 1 e. ZZ
50		zsubcl 9638	. . . . . . . . . . . . . . 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 8386	. . . . . . . . . . . . . . 15 |- 1 =/= ; 1 0
53	19, 20	subeq0i 8570	. . . . . . . . . . . . . . . 16 |- ( ( 1 - ; 1 0 ) = 0 <-> 1 = ; 1 0 )
54	53	necon3bii 2452	. . . . . . . . . . . . . . 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 9638	. . . . . . . . . . . . . . 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 12504	. . . . . . . . . . . . . 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 1378	. . . . . . . . . . . . 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 9722	. . . . . . . . . . . . 13 |- ( k e. NN0 -> (; 1 0 ^ k ) e. CC )
63		negsubdi2 8549	. . . . . . . . . . . . 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 4142	. . . . . . . . . . 11 |- ( k e. NN0 -> ( 1 - ; 1 0 ) || -u ( (; 1 0 ^ k ) - 1 ) )
66		peano2zm 9635	. . . . . . . . . . . . 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 12522	. . . . . . . . . . . 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 12521	. . . . . . . . . . 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 4150	. . . . . . . 8 |- ( k e. NN0 -> ( 3 x. 3 ) || ( (; 1 0 ^ k ) - 1 ) )
75		muldvds1 12530	. . . . . . . . 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 1378	. . . . . . . 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 12547	. . . . . . 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 1379	. . . . . 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 9722	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( F ` k ) e. CC )
84	14	zcnd 9722	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> (; 1 0 ^ k ) e. CC )
85	83, 84	muls1d 8709	. . . . 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 4140	. . . 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 12556	. . 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 9722	. . . 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 12166	. . 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 4140	. 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 12549	. 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 1277	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 1398   e. wcel 2205   =/= wne 2414   class class class wbr 4114   -->wf 5353   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   - cmin 8461   -ucneg 8462   # cap 8873   / cdiv 8966   3c3 9309   9c9 9315   NN0cn0 9516   ZZcz 9597  ;cdc 9730   ...cfz 10364   ^cexp 10927   sum_csu 12066   || cdvds 12501

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-div 8967  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-9 9323  df-n0 9517  df-z 9598  df-dec 9731  df-uz 9875  df-q 9973  df-rp 10008  df-fz 10365  df-fzo 10502  df-seqfrec 10837  df-exp 10928  df-ihash 11167  df-cj 11555  df-re 11556  df-im 11557  df-rsqrt 11711  df-abs 11712  df-clim 11992  df-sumdc 12067  df-dvds 12502

This theorem is referenced by: (None)