Theorem fsumdvds 15030

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

Step	Hyp	Ref	Expression
1		0z 11388	. . . 4 |- 0 e. ZZ
2		dvds0 14997	. . . 4 |- ( 0 e. ZZ -> 0 || 0 )
3	1, 2	mp1i 13	. . 3 |- ( ( ph /\ N = 0 ) -> 0 || 0 )
4		simpr 477	. . 3 |- ( ( ph /\ N = 0 ) -> N = 0 )
5		simplr 792	. . . . . . 7 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> N = 0 )
6		fsumdvds.4	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> N || B )
7	6	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> N || B )
8	5, 7	eqbrtrrd 4677	. . . . . 6 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> 0 || B )
9		fsumdvds.3	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> B e. ZZ )
10	9	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> B e. ZZ )
11		0dvds 15002	. . . . . . 7 |- ( B e. ZZ -> ( 0 || B <-> B = 0 ) )
12	10, 11	syl 17	. . . . . 6 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> ( 0 || B <-> B = 0 ) )
13	8, 12	mpbid 222	. . . . 5 |- ( ( ( ph /\ N = 0 ) /\ k e. A ) -> B = 0 )
14	13	sumeq2dv 14433	. . . 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 481	. . . . . 6 |- ( ( ph /\ N = 0 ) -> A e. Fin )
17	16	olcd 408	. . . . 5 |- ( ( ph /\ N = 0 ) -> ( A C_ ( ZZ>= ` 0 ) \/ A e. Fin ) )
18		sumz 14453	. . . . 5 |- ( ( A C_ ( ZZ>= ` 0 ) \/ A e. Fin ) -> sum_ k e. A 0 = 0 )
19	17, 18	syl 17	. . . 4 |- ( ( ph /\ N = 0 ) -> sum_ k e. A 0 = 0 )
20	14, 19	eqtrd 2656	. . 3 |- ( ( ph /\ N = 0 ) -> sum_ k e. A B = 0 )
21	3, 4, 20	3brtr4d 4685	. 2 |- ( ( ph /\ N = 0 ) -> N || sum_ k e. A B )
22	15	adantr 481	. . . . 5 |- ( ( ph /\ N =/= 0 ) -> A e. Fin )
23		fsumdvds.2	. . . . . . 7 |- ( ph -> N e. ZZ )
24	23	adantr 481	. . . . . 6 |- ( ( ph /\ N =/= 0 ) -> N e. ZZ )
25	24	zcnd 11483	. . . . 5 |- ( ( ph /\ N =/= 0 ) -> N e. CC )
26	9	adantlr 751	. . . . . 6 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> B e. ZZ )
27	26	zcnd 11483	. . . . 5 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> B e. CC )
28		simpr 477	. . . . 5 |- ( ( ph /\ N =/= 0 ) -> N =/= 0 )
29	22, 25, 27, 28	fsumdivc 14518	. . . 4 |- ( ( ph /\ N =/= 0 ) -> ( sum_ k e. A B / N ) = sum_ k e. A ( B / N ) )
30	6	adantlr 751	. . . . . 6 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> N || B )
31	24	adantr 481	. . . . . . 7 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> N e. ZZ )
32		simplr 792	. . . . . . 7 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> N =/= 0 )
33		dvdsval2 14986	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 /\ B e. ZZ ) -> ( N || B <-> ( B / N ) e. ZZ ) )
34	31, 32, 26, 33	syl3anc 1326	. . . . . 6 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> ( N || B <-> ( B / N ) e. ZZ ) )
35	30, 34	mpbid 222	. . . . 5 |- ( ( ( ph /\ N =/= 0 ) /\ k e. A ) -> ( B / N ) e. ZZ )
36	22, 35	fsumzcl 14466	. . . 4 |- ( ( ph /\ N =/= 0 ) -> sum_ k e. A ( B / N ) e. ZZ )
37	29, 36	eqeltrd 2701	. . 3 |- ( ( ph /\ N =/= 0 ) -> ( sum_ k e. A B / N ) e. ZZ )
38	15, 9	fsumzcl 14466	. . . . 5 |- ( ph -> sum_ k e. A B e. ZZ )
39	38	adantr 481	. . . 4 |- ( ( ph /\ N =/= 0 ) -> sum_ k e. A B e. ZZ )
40		dvdsval2 14986	. . . 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 ) )
41	24, 28, 39, 40	syl3anc 1326	. . 3 |- ( ( ph /\ N =/= 0 ) -> ( N || sum_ k e. A B <-> ( sum_ k e. A B / N ) e. ZZ ) )
42	37, 41	mpbird 247	. 2 |- ( ( ph /\ N =/= 0 ) -> N || sum_ k e. A B )
43	21, 42	pm2.61dane 2881	1 |- ( ph -> N || sum_ k e. A B )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   / cdiv 10684   ZZcz 11377   ZZ>=cuz 11687   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-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:  3dvds  15052  3dvdsOLD  15053  sylow1lem3  18015  sylow2alem2  18033  poimirlem26  33435  poimirlem27  33436  etransclem37  40488  etransclem38  40489  etransclem44  40495