Theorem fsumdvds 12374

Description: If every term in a sum is divisible by N, then so is the sum. (Contributed by Mario Carneiro, 17-Jan-2015.)

Hypotheses

Ref	Expression
fsumdvds.1	|- ( ph -> A e. Fin )
fsumdvds.2	|- ( ph -> N e. ZZ )
fsumdvds.3	|- ( ( ph /\ k e. A ) -> B e. ZZ )
fsumdvds.4	|- ( ( ph /\ k e. A ) -> N || B )

Assertion

Ref	Expression
fsumdvds	|- ( ph -> N || sum_ k e. A B )

Distinct variable groups:   A, k   k, N   ph, k

Allowed substitution hint:   B( k)

Proof of Theorem fsumdvds

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0z 9473	. . . 4 |- 0 e. ZZ
2		dvds0 12338	. . . 4 |- ( 0 e. ZZ -> 0 || 0 )
3	1, 2	mp1i 10	. . 3 |- ( ( ph /\ N = 0 ) -> 0 || 0 )
4		simpr 110	. . 3 |- ( ( ph /\ N = 0 ) -> N = 0 )
5		simplr 528	. . . . . . 7 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> N = 0 )
6		fsumdvds.4	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> N || B )
7	6	adantlr 477	. . . . . . 7 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> N || B )
8	5, 7	eqbrtrrd 4107	. . . . . 6 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> 0 || B )
9		fsumdvds.3	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> B e. ZZ )
10	9	adantlr 477	. . . . . . 7 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> B e. ZZ )
11		0dvds 12343	. . . . . . 7 |- ( B e. ZZ -> ( 0 || B <-> B = 0 ) )
12	10, 11	syl 14	. . . . . 6 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> ( 0 || B <-> B = 0 ) )
13	8, 12	mpbid 147	. . . . 5 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> B = 0 )
14	13	sumeq2dv 11900	. . . 4 |- ( ( ph /\ N = 0 ) -> sum_ k e. A B = sum_ k e. A 0 )
15		fsumdvds.1	. . . . . . 7 |- ( ph -> A e. Fin )
16	15	adantr 276	. . . . . 6 |- ( ( ph /\ N = 0 ) -> A e. Fin )
17	16	olcd 739	. . . . 5 |- ( ( ph /\ N = 0 ) -> ( ( 0 e. ZZ /\ A C_ ( ZZ>= ` 0 ) /\ A. j e. ( ZZ>= ` 0 )DECID j e. A ) \/ A e. Fin ) )
18		isumz 11921	. . . . 5 |- ( ( ( 0 e. ZZ /\ A C_ ( ZZ>= ` 0 ) /\ A. j e. ( ZZ>= ` 0 )DECID j e. A ) \/ A e. Fin ) -> sum_ k e. A 0 = 0 )
19	17, 18	syl 14	. . . 4 |- ( ( ph /\ N = 0 ) -> sum_ k e. A 0 = 0 )
20	14, 19	eqtrd 2262	. . 3 |- ( ( ph /\ N = 0 ) -> sum_ k e. A B = 0 )
21	3, 4, 20	3brtr4d 4115	. 2 |- ( ( ph /\ N = 0 ) -> N || sum_ k e. A B )
22	15	adantr 276	. . . . 5 |- ( ( ph /\ N =/= 0 ) -> A e. Fin )
23		fsumdvds.2	. . . . . . 7 |- ( ph -> N e. ZZ )
24	23	adantr 276	. . . . . 6 |- ( ( ph /\ N =/= 0 ) -> N e. ZZ )
25	24	zcnd 9586	. . . . 5 |- ( ( ph /\ N =/= 0 ) -> N e. CC )
26	9	adantlr 477	. . . . . 6 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> B e. ZZ )
27	26	zcnd 9586	. . . . 5 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> B e. CC )
28		zapne 9537	. . . . . . 7 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N # 0 <-> N =/= 0 ) )
29	23, 1, 28	sylancl 413	. . . . . 6 |- ( ph -> ( N # 0 <-> N =/= 0 ) )
30	29	biimpar 297	. . . . 5 |- ( ( ph /\ N =/= 0 ) -> N # 0 )
31	22, 25, 27, 30	fsumdivapc 11982	. . . 4 |- ( ( ph /\ N =/= 0 ) -> ( sum_ k e. A B / N ) = sum_ k e. A ( B / N ) )
32	6	adantlr 477	. . . . . 6 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> N || B )
33	24	adantr 276	. . . . . . 7 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> N e. ZZ )
34		simplr 528	. . . . . . 7 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> N =/= 0 )
35		dvdsval2 12322	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 /\ B e. ZZ ) -> ( N || B <-> ( B / N ) e. ZZ ) )
36	33, 34, 26, 35	syl3anc 1271	. . . . . 6 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> ( N || B <-> ( B / N ) e. ZZ ) )
37	32, 36	mpbid 147	. . . . 5 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> ( B / N ) e. ZZ )
38	22, 37	fsumzcl 11934	. . . 4 |- ( ( ph /\ N =/= 0 ) -> sum_ k e. A ( B / N ) e. ZZ )
39	31, 38	eqeltrd 2306	. . 3 |- ( ( ph /\ N =/= 0 ) -> ( sum_ k e. A B / N ) e. ZZ )
40		simpr 110	. . . 4 |- ( ( ph /\ N =/= 0 ) -> N =/= 0 )
41	15, 9	fsumzcl 11934	. . . . 5 |- ( ph -> sum_ k e. A B e. ZZ )
42	41	adantr 276	. . . 4 |- ( ( ph /\ N =/= 0 ) -> sum_ k e. A B e. ZZ )
43		dvdsval2 12322	. . . 4 |- ( ( N e. ZZ /\ N =/= 0 /\ sum_ k e. A B e. ZZ ) -> ( N || sum_ k e. A B <-> ( sum_ k e. A B / N ) e. ZZ ) )
44	24, 40, 42, 43	syl3anc 1271	. . 3 |- ( ( ph /\ N =/= 0 ) -> ( N || sum_ k e. A B <-> ( sum_ k e. A B / N ) e. ZZ ) )
45	39, 44	mpbird 167	. 2 |- ( ( ph /\ N =/= 0 ) -> N || sum_ k e. A B )
46		zdceq 9538	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
47	23, 1, 46	sylancl 413	. . 3 |- ( ph -> DECID N = 0 )
48		dcne 2411	. . 3 |- (DECID N = 0 <-> ( N = 0 \/ N =/= 0 ) )
49	47, 48	sylib 122	. 2 |- ( ph -> ( N = 0 \/ N =/= 0 ) )
50	21, 45, 49	mpjaodan 803	1 |- ( ph -> N || sum_ k e. A B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   C_ wss 3197   class class class wbr 4083   ` cfv 5321  (class class class)co 6010   Fincfn 6900   0cc0 8015   # cap 8744   / cdiv 8835   ZZcz 9462   ZZ>=cuz 9738   sum_csu 11885   || cdvds 12319

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-isom 5330  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-frec 6548  df-1o 6573  df-oadd 6577  df-er 6693  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-q 9832  df-rp 9867  df-fz 10222  df-fzo 10356  df-seqfrec 10687  df-exp 10778  df-ihash 11015  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-clim 11811  df-sumdc 11886  df-dvds 12320

This theorem is referenced by:  3dvds  12396