Theorem fsumdvds 12339

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 9445	. . . 4 |- 0 e. ZZ
2		dvds0 12303	. . . 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 4106	. . . . . 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 12308	. . . . . . 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 11865	. . . 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 11886	. . . . 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 4114	. 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 9558	. . . . 5 |- ( ( ph /\ N =/= 0 ) -> N e. CC )
26	9	adantlr 477	. . . . . 6 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> B e. ZZ )
27	26	zcnd 9558	. . . . 5 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> B e. CC )
28		zapne 9509	. . . . . . 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 11947	. . . 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 12287	. . . . . . 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 11899	. . . 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 11899	. . . . 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 12287	. . . 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 9510	. . . 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 4082   ` cfv 5314  (class class class)co 5994   Fincfn 6877   0cc0 7987   # cap 8716   / cdiv 8807   ZZcz 9434   ZZ>=cuz 9710   sum_csu 11850   || cdvds 12284

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-isom 5323  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-frec 6527  df-1o 6552  df-oadd 6556  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-fz 10193  df-fzo 10327  df-seqfrec 10657  df-exp 10748  df-ihash 10985  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-clim 11776  df-sumdc 11851  df-dvds 12285

This theorem is referenced by:  3dvds  12361