Theorem fsumdvds 12592

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 9638	. . . 4 |- 0 e. ZZ
2		dvds0 12556	. . . 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 533	. . . . . . 7 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> N = 0 )
6		fsumdvds.4	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> N || B )
7	6	adantlr 481	. . . . . . 7 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> N || B )
8	5, 7	eqbrtrrd 4152	. . . . . 6 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> 0 || B )
9		fsumdvds.3	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> B e. ZZ )
10	9	adantlr 481	. . . . . . 7 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> B e. ZZ )
11		0dvds 12561	. . . . . . 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 12117	. . . 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 746	. . . . 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 12139	. . . . 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 2271	. . 3 |- ( ( ph /\ N = 0 ) -> sum_ k e. A B = 0 )
21	3, 4, 20	3brtr4d 4160	. 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 9752	. . . . 5 |- ( ( ph /\ N =/= 0 ) -> N e. CC )
26	9	adantlr 481	. . . . . 6 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> B e. ZZ )
27	26	zcnd 9752	. . . . 5 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> B e. CC )
28		zapne 9702	. . . . . . 7 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N # 0 <-> N =/= 0 ) )
29	23, 1, 28	sylancl 417	. . . . . 6 |- ( ph -> ( N # 0 <-> N =/= 0 ) )
30	29	biimpar 297	. . . . 5 |- ( ( ph /\ N =/= 0 ) -> N # 0 )
31	22, 25, 27, 30	fsumdivapc 12200	. . . 4 |- ( ( ph /\ N =/= 0 ) -> ( sum_ k e. A B / N ) = sum_ k e. A ( B / N ) )
32	6	adantlr 481	. . . . . 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 533	. . . . . . 7 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> N =/= 0 )
35		dvdsval2 12540	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 /\ B e. ZZ ) -> ( N || B <-> ( B / N ) e. ZZ ) )
36	33, 34, 26, 35	syl3anc 1278	. . . . . 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 12152	. . . 4 |- ( ( ph /\ N =/= 0 ) -> sum_ k e. A ( B / N ) e. ZZ )
39	31, 38	eqeltrd 2315	. . 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 12152	. . . . 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 12540	. . . 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 1278	. . 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 9703	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
47	23, 1, 46	sylancl 417	. . 3 |- ( ph -> DECID N = 0 )
48		dcne 2431	. . 3 |- (DECID N = 0 <-> ( N = 0 \/ N =/= 0 ) )
49	47, 48	sylib 122	. 2 |- ( ph -> ( N = 0 \/ N =/= 0 ) )
50	21, 45, 49	mpjaodan 810	1 |- ( ph -> N || sum_ k e. A B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   C_ wss 3220   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   Fincfn 7016   0cc0 8173   # cap 8903   / cdiv 8996   ZZcz 9627   ZZ>=cuz 9904   sum_csu 12102   || cdvds 12537

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-sumdc 12103  df-dvds 12538

This theorem is referenced by:  3dvds  12614