Theorem fsumdvds 12464

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 9533	. . . 4 |- 0 e. ZZ
2		dvds0 12428	. . . 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 529	. . . . . . 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 4117	. . . . . 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 12433	. . . . . . 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 11989	. . . 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 742	. . . . 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 12011	. . . . 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 2264	. . 3 |- ( ( ph /\ N = 0 ) -> sum_ k e. A B = 0 )
21	3, 4, 20	3brtr4d 4125	. 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 9646	. . . . 5 |- ( ( ph /\ N =/= 0 ) -> N e. CC )
26	9	adantlr 477	. . . . . 6 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> B e. ZZ )
27	26	zcnd 9646	. . . . 5 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> B e. CC )
28		zapne 9597	. . . . . . 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 12072	. . . 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 529	. . . . . . 7 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> N =/= 0 )
35		dvdsval2 12412	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 /\ B e. ZZ ) -> ( N || B <-> ( B / N ) e. ZZ ) )
36	33, 34, 26, 35	syl3anc 1274	. . . . . 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 12024	. . . 4 |- ( ( ph /\ N =/= 0 ) -> sum_ k e. A ( B / N ) e. ZZ )
39	31, 38	eqeltrd 2308	. . 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 12024	. . . . 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 12412	. . . 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 1274	. . 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 9598	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
47	23, 1, 46	sylancl 413	. . 3 |- ( ph -> DECID N = 0 )
48		dcne 2414	. . 3 |- (DECID N = 0 <-> ( N = 0 \/ N =/= 0 ) )
49	47, 48	sylib 122	. 2 |- ( ph -> ( N = 0 \/ N =/= 0 ) )
50	21, 45, 49	mpjaodan 806	1 |- ( ph -> N || sum_ k e. A B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2202   =/= wne 2403   A.wral 2511   C_ wss 3201   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   Fincfn 6952   0cc0 8075   # cap 8804   / cdiv 8895   ZZcz 9522   ZZ>=cuz 9798   sum_csu 11974   || cdvds 12409

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-n0 9446  df-z 9523  df-uz 9799  df-q 9897  df-rp 9932  df-fz 10287  df-fzo 10421  df-seqfrec 10754  df-exp 10845  df-ihash 11082  df-cj 11463  df-re 11464  df-im 11465  df-rsqrt 11619  df-abs 11620  df-clim 11900  df-sumdc 11975  df-dvds 12410

This theorem is referenced by:  3dvds  12486