Theorem 3dvds 12219

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 9408	. . 3 |- 3 e. ZZ
2	1	a1i 9	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 3 e. ZZ )
3		0zd 9391	. . . 4 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 0 e. ZZ )
4		nn0z 9399	. . . . 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 10583	. . 3 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> ( 0 ... N ) e. Fin )
7		ffvelcdm 5720	. . . . 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 9526	. . . . . 6 |- ; 1 0 e. NN
10	9	nnzi 9400	. . . . 5 |- ; 1 0 e. ZZ
11		elfznn0 10243	. . . . . 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 10706	. . . . 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 9508	. . 3 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( F ` k ) x. (; 1 0 ^ k ) ) e. ZZ )
16	6, 15	fsumzcl 11757	. 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 11757	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> sum_ k e. ( 0 ... N ) ( F ` k ) e. ZZ )
18	15, 8	zsubcld 9507	. . . 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 8025	. . . . . . . . . . . 12 |- 1 e. CC
20	9	nncni 9053	. . . . . . . . . . . 12 |- ; 1 0 e. CC
21	19, 20	negsubdi2i 8365	. . . . . . . . . . 11 |- -u ( 1 - ; 1 0 ) = (; 1 0 - 1 )
22		9p1e10 9513	. . . . . . . . . . . . 13 |- ( 9 + 1 ) = ; 1 0
23	22	eqcomi 2210	. . . . . . . . . . . 12 |- ; 1 0 = ( 9 + 1 )
24	23	oveq1i 5961	. . . . . . . . . . 11 |- (; 1 0 - 1 ) = ( ( 9 + 1 ) - 1 )
25		9cn 9131	. . . . . . . . . . . 12 |- 9 e. CC
26	25, 19	pncan3oi 8295	. . . . . . . . . . 11 |- ( ( 9 + 1 ) - 1 ) = 9
27	21, 24, 26	3eqtri 2231	. . . . . . . . . 10 |- -u ( 1 - ; 1 0 ) = 9
28		3t3e9 9201	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
29	27, 28	eqtr4i 2230	. . . . . . . . 9 |- -u ( 1 - ; 1 0 ) = ( 3 x. 3 )
30	20	a1i 9	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> ; 1 0 e. CC )
31		1re 8078	. . . . . . . . . . . . . . . . 17 |- 1 e. RR
32		10re 9529	. . . . . . . . . . . . . . . . 17 |- ; 1 0 e. RR
33		1lt10 9649	. . . . . . . . . . . . . . . . 17 |- 1 < ; 1 0
34	31, 32, 33	gtapii 8714	. . . . . . . . . . . . . . . 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 11862	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> sum_ j e. ( 0 ... ( k - 1 ) ) (; 1 0 ^ j ) = ( ( 1 - (; 1 0 ^ k ) ) / ( 1 - ; 1 0 ) ) )
38		0zd 9391	. . . . . . . . . . . . . . . 16 |- ( k e. NN0 -> 0 e. ZZ )
39		nn0z 9399	. . . . . . . . . . . . . . . . 17 |- ( k e. NN0 -> k e. ZZ )
40		peano2zm 9417	. . . . . . . . . . . . . . . . 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 10583	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> ( 0 ... ( k - 1 ) ) e. Fin )
43		elfznn0 10243	. . . . . . . . . . . . . . . . 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 10706	. . . . . . . . . . . . . . . 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 11757	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> sum_ j e. ( 0 ... ( k - 1 ) ) (; 1 0 ^ j ) e. ZZ )
48	37, 47	eqeltrrd 2284	. . . . . . . . . . . . 13 |- ( k e. NN0 -> ( ( 1 - (; 1 0 ^ k ) ) / ( 1 - ; 1 0 ) ) e. ZZ )
49		1z 9405	. . . . . . . . . . . . . . 15 |- 1 e. ZZ
50		zsubcl 9420	. . . . . . . . . . . . . . 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 8176	. . . . . . . . . . . . . . 15 |- 1 =/= ; 1 0
53	19, 20	subeq0i 8359	. . . . . . . . . . . . . . . 16 |- ( ( 1 - ; 1 0 ) = 0 <-> 1 = ; 1 0 )
54	53	necon3bii 2415	. . . . . . . . . . . . . . 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 9420	. . . . . . . . . . . . . . 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 12145	. . . . . . . . . . . . . 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 1354	. . . . . . . . . . . . 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 9503	. . . . . . . . . . . . 13 |- ( k e. NN0 -> (; 1 0 ^ k ) e. CC )
63		negsubdi2 8338	. . . . . . . . . . . . 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 4075	. . . . . . . . . . 11 |- ( k e. NN0 -> ( 1 - ; 1 0 ) || -u ( (; 1 0 ^ k ) - 1 ) )
66		peano2zm 9417	. . . . . . . . . . . . 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 12163	. . . . . . . . . . . 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 12162	. . . . . . . . . . 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 4083	. . . . . . . 8 |- ( k e. NN0 -> ( 3 x. 3 ) || ( (; 1 0 ^ k ) - 1 ) )
75		muldvds1 12171	. . . . . . . . 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 1354	. . . . . . . 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 12188	. . . . . . 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 1355	. . . . . 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 9503	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( F ` k ) e. CC )
84	14	zcnd 9503	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> (; 1 0 ^ k ) e. CC )
85	83, 84	muls1d 8497	. . . . 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 4073	. . . 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 12197	. . 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 9503	. . . 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 11807	. . 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 4073	. 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 12190	. 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 1253	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 1373   e. wcel 2177   =/= wne 2377   class class class wbr 4047   -->wf 5272   ` cfv 5276  (class class class)co 5951   CCcc 7930   0cc0 7932   1c1 7933   + caddc 7935   x. cmul 7937   - cmin 8250   -ucneg 8251   # cap 8661   / cdiv 8752   3c3 9095   9c9 9101   NN0cn0 9302   ZZcz 9379  ;cdc 9511   ...cfz 10137   ^cexp 10690   sum_csu 11708   || cdvds 12142

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-frec 6484  df-1o 6509  df-oadd 6513  df-er 6627  df-en 6835  df-dom 6836  df-fin 6837  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-9 9109  df-n0 9303  df-z 9380  df-dec 9512  df-uz 9656  df-q 9748  df-rp 9783  df-fz 10138  df-fzo 10272  df-seqfrec 10600  df-exp 10691  df-ihash 10928  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-clim 11634  df-sumdc 11709  df-dvds 12143

This theorem is referenced by: (None)