Theorem 3dvdsOLD 15053

Description: Obsolete version of 3dvds 15052 as of 8-Sep-2021. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
3dvdsOLD	|- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> ( 3 || sum_ k e. ( 0 ... N ) ( ( F ` k ) x. ( 10 ^ k ) ) <-> 3 || sum_ k e. ( 0 ... N ) ( F ` k ) ) )

Distinct variable groups:   k, F   k, N

Proof of Theorem 3dvdsOLD

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3z 11410	. . 3 |- 3 e. ZZ
2	1	a1i 11	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 3 e. ZZ )
3		fzfid 12772	. . 3 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> ( 0 ... N ) e. Fin )
4		ffvelrn 6357	. . . . 5 |- ( ( F : ( 0 ... N ) --> ZZ /\ k e. ( 0 ... N ) ) -> ( F ` k ) e. ZZ )
5	4	adantll 750	. . . 4 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( F ` k ) e. ZZ )
6		10nnOLD 11193	. . . . . 6 |- 10 e. NN
7	6	nnzi 11401	. . . . 5 |- 10 e. ZZ
8		elfznn0 12433	. . . . . 6 |- ( k e. ( 0 ... N ) -> k e. NN0 )
9	8	adantl 482	. . . . 5 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> k e. NN0 )
10		zexpcl 12875	. . . . 5 |- ( ( 10 e. ZZ /\ k e. NN0 ) -> ( 10 ^ k ) e. ZZ )
11	7, 9, 10	sylancr 695	. . . 4 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( 10 ^ k ) e. ZZ )
12	5, 11	zmulcld 11488	. . 3 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( F ` k ) x. ( 10 ^ k ) ) e. ZZ )
13	3, 12	fsumzcl 14466	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> sum_ k e. ( 0 ... N ) ( ( F ` k ) x. ( 10 ^ k ) ) e. ZZ )
14	3, 5	fsumzcl 14466	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> sum_ k e. ( 0 ... N ) ( F ` k ) e. ZZ )
15	12, 5	zsubcld 11487	. . . 4 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( ( F ` k ) x. ( 10 ^ k ) ) - ( F ` k ) ) e. ZZ )
16		ax-1cn 9994	. . . . . . . . . . . 12 |- 1 e. CC
17	6	nncni 11030	. . . . . . . . . . . 12 |- 10 e. CC
18	16, 17	negsubdi2i 10367	. . . . . . . . . . 11 |- -u ( 1 - 10 ) = ( 10 - 1 )
19		df-10OLD 11087	. . . . . . . . . . . 12 |- 10 = ( 9 + 1 )
20	19	oveq1i 6660	. . . . . . . . . . 11 |- ( 10 - 1 ) = ( ( 9 + 1 ) - 1 )
21		9cn 11108	. . . . . . . . . . . 12 |- 9 e. CC
22	21, 16	pncan3oi 10297	. . . . . . . . . . 11 |- ( ( 9 + 1 ) - 1 ) = 9
23	18, 20, 22	3eqtri 2648	. . . . . . . . . 10 |- -u ( 1 - 10 ) = 9
24		3t3e9 11180	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
25	23, 24	eqtr4i 2647	. . . . . . . . 9 |- -u ( 1 - 10 ) = ( 3 x. 3 )
26	17	a1i 11	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> 10 e. CC )
27		1re 10039	. . . . . . . . . . . . . . . . 17 |- 1 e. RR
28		1lt10OLD 11238	. . . . . . . . . . . . . . . . 17 |- 1 < 10
29	27, 28	gtneii 10149	. . . . . . . . . . . . . . . 16 |- 10 =/= 1
30	29	a1i 11	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> 10 =/= 1 )
31		id 22	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> k e. NN0 )
32	26, 30, 31	geoser 14599	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> sum_ j e. ( 0 ... ( k - 1 ) ) ( 10 ^ j ) = ( ( 1 - ( 10 ^ k ) ) / ( 1 - 10 ) ) )
33		fzfid 12772	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> ( 0 ... ( k - 1 ) ) e. Fin )
34		elfznn0 12433	. . . . . . . . . . . . . . . . 17 |- ( j e. ( 0 ... ( k - 1 ) ) -> j e. NN0 )
35	34	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( k e. NN0 /\ j e. ( 0 ... ( k - 1 ) ) ) -> j e. NN0 )
36		zexpcl 12875	. . . . . . . . . . . . . . . 16 |- ( ( 10 e. ZZ /\ j e. NN0 ) -> ( 10 ^ j ) e. ZZ )
37	7, 35, 36	sylancr 695	. . . . . . . . . . . . . . 15 |- ( ( k e. NN0 /\ j e. ( 0 ... ( k - 1 ) ) ) -> ( 10 ^ j ) e. ZZ )
38	33, 37	fsumzcl 14466	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> sum_ j e. ( 0 ... ( k - 1 ) ) ( 10 ^ j ) e. ZZ )
39	32, 38	eqeltrrd 2702	. . . . . . . . . . . . 13 |- ( k e. NN0 -> ( ( 1 - ( 10 ^ k ) ) / ( 1 - 10 ) ) e. ZZ )
40		1z 11407	. . . . . . . . . . . . . . . 16 |- 1 e. ZZ
41		zsubcl 11419	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. ZZ /\ 10 e. ZZ ) -> ( 1 - 10 ) e. ZZ )
42	40, 7, 41	mp2an 708	. . . . . . . . . . . . . . 15 |- ( 1 - 10 ) e. ZZ
43	42	a1i 11	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> ( 1 - 10 ) e. ZZ )
44	27, 28	ltneii 10150	. . . . . . . . . . . . . . . 16 |- 1 =/= 10
45	16, 17	subeq0i 10361	. . . . . . . . . . . . . . . . 17 |- ( ( 1 - 10 ) = 0 <-> 1 = 10 )
46	45	necon3bii 2846	. . . . . . . . . . . . . . . 16 |- ( ( 1 - 10 ) =/= 0 <-> 1 =/= 10 )
47	44, 46	mpbir 221	. . . . . . . . . . . . . . 15 |- ( 1 - 10 ) =/= 0
48	47	a1i 11	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> ( 1 - 10 ) =/= 0 )
49	7, 31, 10	sylancr 695	. . . . . . . . . . . . . . 15 |- ( k e. NN0 -> ( 10 ^ k ) e. ZZ )
50		zsubcl 11419	. . . . . . . . . . . . . . 15 |- ( ( 1 e. ZZ /\ ( 10 ^ k ) e. ZZ ) -> ( 1 - ( 10 ^ k ) ) e. ZZ )
51	40, 49, 50	sylancr 695	. . . . . . . . . . . . . 14 |- ( k e. NN0 -> ( 1 - ( 10 ^ k ) ) e. ZZ )
52		dvdsval2 14986	. . . . . . . . . . . . . 14 |- ( ( ( 1 - 10 ) e. ZZ /\ ( 1 - 10 ) =/= 0 /\ ( 1 - ( 10 ^ k ) ) e. ZZ ) -> ( ( 1 - 10 ) || ( 1 - ( 10 ^ k ) ) <-> ( ( 1 - ( 10 ^ k ) ) / ( 1 - 10 ) ) e. ZZ ) )
53	43, 48, 51, 52	syl3anc 1326	. . . . . . . . . . . . 13 |- ( k e. NN0 -> ( ( 1 - 10 ) || ( 1 - ( 10 ^ k ) ) <-> ( ( 1 - ( 10 ^ k ) ) / ( 1 - 10 ) ) e. ZZ ) )
54	39, 53	mpbird 247	. . . . . . . . . . . 12 |- ( k e. NN0 -> ( 1 - 10 ) || ( 1 - ( 10 ^ k ) ) )
55	49	zcnd 11483	. . . . . . . . . . . . 13 |- ( k e. NN0 -> ( 10 ^ k ) e. CC )
56		negsubdi2 10340	. . . . . . . . . . . . 13 |- ( ( ( 10 ^ k ) e. CC /\ 1 e. CC ) -> -u ( ( 10 ^ k ) - 1 ) = ( 1 - ( 10 ^ k ) ) )
57	55, 16, 56	sylancl 694	. . . . . . . . . . . 12 |- ( k e. NN0 -> -u ( ( 10 ^ k ) - 1 ) = ( 1 - ( 10 ^ k ) ) )
58	54, 57	breqtrrd 4681	. . . . . . . . . . 11 |- ( k e. NN0 -> ( 1 - 10 ) || -u ( ( 10 ^ k ) - 1 ) )
59		peano2zm 11420	. . . . . . . . . . . . 13 |- ( ( 10 ^ k ) e. ZZ -> ( ( 10 ^ k ) - 1 ) e. ZZ )
60	49, 59	syl 17	. . . . . . . . . . . 12 |- ( k e. NN0 -> ( ( 10 ^ k ) - 1 ) e. ZZ )
61		dvdsnegb 14999	. . . . . . . . . . . 12 |- ( ( ( 1 - 10 ) e. ZZ /\ ( ( 10 ^ k ) - 1 ) e. ZZ ) -> ( ( 1 - 10 ) || ( ( 10 ^ k ) - 1 ) <-> ( 1 - 10 ) || -u ( ( 10 ^ k ) - 1 ) ) )
62	42, 60, 61	sylancr 695	. . . . . . . . . . 11 |- ( k e. NN0 -> ( ( 1 - 10 ) || ( ( 10 ^ k ) - 1 ) <-> ( 1 - 10 ) || -u ( ( 10 ^ k ) - 1 ) ) )
63	58, 62	mpbird 247	. . . . . . . . . 10 |- ( k e. NN0 -> ( 1 - 10 ) || ( ( 10 ^ k ) - 1 ) )
64		negdvdsb 14998	. . . . . . . . . . 11 |- ( ( ( 1 - 10 ) e. ZZ /\ ( ( 10 ^ k ) - 1 ) e. ZZ ) -> ( ( 1 - 10 ) || ( ( 10 ^ k ) - 1 ) <-> -u ( 1 - 10 ) || ( ( 10 ^ k ) - 1 ) ) )
65	42, 60, 64	sylancr 695	. . . . . . . . . 10 |- ( k e. NN0 -> ( ( 1 - 10 ) || ( ( 10 ^ k ) - 1 ) <-> -u ( 1 - 10 ) || ( ( 10 ^ k ) - 1 ) ) )
66	63, 65	mpbid 222	. . . . . . . . 9 |- ( k e. NN0 -> -u ( 1 - 10 ) || ( ( 10 ^ k ) - 1 ) )
67	25, 66	syl5eqbrr 4689	. . . . . . . 8 |- ( k e. NN0 -> ( 3 x. 3 ) || ( ( 10 ^ k ) - 1 ) )
68	1	a1i 11	. . . . . . . . 9 |- ( k e. NN0 -> 3 e. ZZ )
69		muldvds1 15006	. . . . . . . . 9 |- ( ( 3 e. ZZ /\ 3 e. ZZ /\ ( ( 10 ^ k ) - 1 ) e. ZZ ) -> ( ( 3 x. 3 ) || ( ( 10 ^ k ) - 1 ) -> 3 || ( ( 10 ^ k ) - 1 ) ) )
70	68, 68, 60, 69	syl3anc 1326	. . . . . . . 8 |- ( k e. NN0 -> ( ( 3 x. 3 ) || ( ( 10 ^ k ) - 1 ) -> 3 || ( ( 10 ^ k ) - 1 ) ) )
71	67, 70	mpd 15	. . . . . . 7 |- ( k e. NN0 -> 3 || ( ( 10 ^ k ) - 1 ) )
72	9, 71	syl 17	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> 3 || ( ( 10 ^ k ) - 1 ) )
73	1	a1i 11	. . . . . . 7 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> 3 e. ZZ )
74	11, 59	syl 17	. . . . . . 7 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( 10 ^ k ) - 1 ) e. ZZ )
75		dvdsmultr2 15021	. . . . . . 7 |- ( ( 3 e. ZZ /\ ( F ` k ) e. ZZ /\ ( ( 10 ^ k ) - 1 ) e. ZZ ) -> ( 3 || ( ( 10 ^ k ) - 1 ) -> 3 || ( ( F ` k ) x. ( ( 10 ^ k ) - 1 ) ) ) )
76	73, 5, 74, 75	syl3anc 1326	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( 3 || ( ( 10 ^ k ) - 1 ) -> 3 || ( ( F ` k ) x. ( ( 10 ^ k ) - 1 ) ) ) )
77	72, 76	mpd 15	. . . . 5 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> 3 || ( ( F ` k ) x. ( ( 10 ^ k ) - 1 ) ) )
78	5	zcnd 11483	. . . . . . 7 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( F ` k ) e. CC )
79	11	zcnd 11483	. . . . . . 7 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( 10 ^ k ) e. CC )
80	16	a1i 11	. . . . . . 7 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> 1 e. CC )
81	78, 79, 80	subdid 10486	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( F ` k ) x. ( ( 10 ^ k ) - 1 ) ) = ( ( ( F ` k ) x. ( 10 ^ k ) ) - ( ( F ` k ) x. 1 ) ) )
82	78	mulid1d 10057	. . . . . . 7 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( F ` k ) x. 1 ) = ( F ` k ) )
83	82	oveq2d 6666	. . . . . 6 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( ( F ` k ) x. ( 10 ^ k ) ) - ( ( F ` k ) x. 1 ) ) = ( ( ( F ` k ) x. ( 10 ^ k ) ) - ( F ` k ) ) )
84	81, 83	eqtrd 2656	. . . . 5 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( F ` k ) x. ( ( 10 ^ k ) - 1 ) ) = ( ( ( F ` k ) x. ( 10 ^ k ) ) - ( F ` k ) ) )
85	77, 84	breqtrd 4679	. . . 4 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> 3 || ( ( ( F ` k ) x. ( 10 ^ k ) ) - ( F ` k ) ) )
86	3, 2, 15, 85	fsumdvds 15030	. . 3 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 3 || sum_ k e. ( 0 ... N ) ( ( ( F ` k ) x. ( 10 ^ k ) ) - ( F ` k ) ) )
87	12	zcnd 11483	. . . 4 |- ( ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) /\ k e. ( 0 ... N ) ) -> ( ( F ` k ) x. ( 10 ^ k ) ) e. CC )
88	3, 87, 78	fsumsub 14520	. . 3 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> sum_ k e. ( 0 ... N ) ( ( ( F ` k ) x. ( 10 ^ k ) ) - ( F ` k ) ) = ( sum_ k e. ( 0 ... N ) ( ( F ` k ) x. ( 10 ^ k ) ) - sum_ k e. ( 0 ... N ) ( F ` k ) ) )
89	86, 88	breqtrd 4679	. 2 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> 3 || ( sum_ k e. ( 0 ... N ) ( ( F ` k ) x. ( 10 ^ k ) ) - sum_ k e. ( 0 ... N ) ( F ` k ) ) )
90		dvdssub2 15023	. 2 |- ( ( ( 3 e. ZZ /\ sum_ k e. ( 0 ... N ) ( ( F ` k ) x. ( 10 ^ k ) ) e. ZZ /\ sum_ k e. ( 0 ... N ) ( F ` k ) e. ZZ ) /\ 3 || ( sum_ k e. ( 0 ... N ) ( ( F ` k ) x. ( 10 ^ k ) ) - sum_ k e. ( 0 ... N ) ( F ` k ) ) ) -> ( 3 || sum_ k e. ( 0 ... N ) ( ( F ` k ) x. ( 10 ^ k ) ) <-> 3 || sum_ k e. ( 0 ... N ) ( F ` k ) ) )
91	2, 13, 14, 89, 90	syl31anc 1329	1 |- ( ( N e. NN0 /\ F : ( 0 ... N ) --> ZZ ) -> ( 3 || sum_ k e. ( 0 ... N ) ( ( F ` k ) x. ( 10 ^ k ) ) <-> 3 || sum_ k e. ( 0 ... N ) ( F ` k ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   3c3 11071   9c9 11077   10c10 11078   NN0cn0 11292   ZZcz 11377   ...cfz 12326   ^cexp 12860   sum_csu 14416   || cdvds 14983

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-tru 1486  df-fal 1489  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-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-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-10OLD 11087  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  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

This theorem is referenced by: (None)