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Theorem fsumdvds 13689
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
StepHypRef Expression
1 0z 10763 . . . 4  |-  0  e.  ZZ
2 dvds0 13661 . . . 4  |-  ( 0  e.  ZZ  ->  0  ||  0 )
31, 2mp1i 12 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  0  ||  0 )
4 simpr 461 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  N  =  0 )
5 simplr 754 . . . . . . 7  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  N  =  0 )
6 fsumdvds.4 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  N  ||  B )
76adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  N  ||  B )
85, 7eqbrtrrd 4417 . . . . . 6  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  0  ||  B )
9 fsumdvds.3 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ZZ )
109adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  B  e.  ZZ )
11 0dvds 13666 . . . . . . 7  |-  ( B  e.  ZZ  ->  (
0  ||  B  <->  B  = 
0 ) )
1210, 11syl 16 . . . . . 6  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  ( 0  ||  B  <->  B  =  0 ) )
138, 12mpbid 210 . . . . 5  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  B  =  0 )
1413sumeq2dv 13293 . . . 4  |-  ( (
ph  /\  N  = 
0 )  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  0 )
15 fsumdvds.1 . . . . . . 7  |-  ( ph  ->  A  e.  Fin )
1615adantr 465 . . . . . 6  |-  ( (
ph  /\  N  = 
0 )  ->  A  e.  Fin )
1716olcd 393 . . . . 5  |-  ( (
ph  /\  N  = 
0 )  ->  ( A  C_  ( ZZ>= `  0
)  \/  A  e. 
Fin ) )
18 sumz 13312 . . . . 5  |-  ( ( A  C_  ( ZZ>= ` 
0 )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  = 
0 )
1917, 18syl 16 . . . 4  |-  ( (
ph  /\  N  = 
0 )  ->  sum_ k  e.  A  0  = 
0 )
2014, 19eqtrd 2493 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  sum_ k  e.  A  B  = 
0 )
213, 4, 203brtr4d 4425 . 2  |-  ( (
ph  /\  N  = 
0 )  ->  N  || 
sum_ k  e.  A  B )
2215adantr 465 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  A  e.  Fin )
23 fsumdvds.2 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
2423adantr 465 . . . . . 6  |-  ( (
ph  /\  N  =/=  0 )  ->  N  e.  ZZ )
2524zcnd 10854 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  N  e.  CC )
269adantlr 714 . . . . . 6  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  B  e.  ZZ )
2726zcnd 10854 . . . . 5  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  B  e.  CC )
28 simpr 461 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  N  =/=  0 )
2922, 25, 27, 28fsumdivc 13366 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  ( sum_ k  e.  A  B  /  N )  =  sum_ k  e.  A  ( B  /  N ) )
306adantlr 714 . . . . . 6  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  N  ||  B )
3124adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  N  e.  ZZ )
32 simplr 754 . . . . . . 7  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  N  =/=  0 )
33 dvdsval2 13651 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  B  e.  ZZ )  ->  ( N  ||  B  <->  ( B  /  N )  e.  ZZ ) )
3431, 32, 26, 33syl3anc 1219 . . . . . 6  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  ( N  ||  B  <->  ( B  /  N )  e.  ZZ ) )
3530, 34mpbid 210 . . . . 5  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  ( B  /  N )  e.  ZZ )
3622, 35fsumzcl 13325 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  sum_ k  e.  A  ( B  /  N )  e.  ZZ )
3729, 36eqeltrd 2540 . . 3  |-  ( (
ph  /\  N  =/=  0 )  ->  ( sum_ k  e.  A  B  /  N )  e.  ZZ )
3815, 9fsumzcl 13325 . . . . 5  |-  ( ph  -> 
sum_ k  e.  A  B  e.  ZZ )
3938adantr 465 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  sum_ k  e.  A  B  e.  ZZ )
40 dvdsval2 13651 . . . 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 ) )
4124, 28, 39, 40syl3anc 1219 . . 3  |-  ( (
ph  /\  N  =/=  0 )  ->  ( N  ||  sum_ k  e.  A  B 
<->  ( sum_ k  e.  A  B  /  N )  e.  ZZ ) )
4237, 41mpbird 232 . 2  |-  ( (
ph  /\  N  =/=  0 )  ->  N  || 
sum_ k  e.  A  B )
4321, 42pm2.61dane 2767 1  |-  ( ph  ->  N  ||  sum_ k  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645    C_ wss 3431   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Fincfn 7415   0cc0 9388    / cdiv 10099   ZZcz 10752   ZZ>=cuz 10967   sum_csu 13276    || cdivides 13648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fz 11550  df-fzo 11661  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-sum 13277  df-dvds 13649
This theorem is referenced by:  3dvds  13709  sylow1lem3  16215  sylow2alem2  16233
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