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Theorem fsumdvds 12893
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 10293 . . . 4  |-  0  e.  ZZ
2 dvds0 12865 . . . 4  |-  ( 0  e.  ZZ  ->  0  ||  0 )
31, 2mp1i 12 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  0  ||  0 )
4 simpr 448 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  N  =  0 )
5 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  N  =  0 )
6 fsumdvds.4 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  N  ||  B )
76adantlr 696 . . . . . . 7  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  N  ||  B )
85, 7eqbrtrrd 4234 . . . . . 6  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  0  ||  B )
9 fsumdvds.3 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ZZ )
109adantlr 696 . . . . . . 7  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  B  e.  ZZ )
11 0dvds 12870 . . . . . . 7  |-  ( B  e.  ZZ  ->  (
0  ||  B  <->  B  = 
0 ) )
1210, 11syl 16 . . . . . 6  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  ( 0  ||  B  <->  B  =  0 ) )
138, 12mpbid 202 . . . . 5  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  B  =  0 )
1413sumeq2dv 12497 . . . 4  |-  ( (
ph  /\  N  = 
0 )  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  0 )
15 fsumdvds.1 . . . . . . 7  |-  ( ph  ->  A  e.  Fin )
1615adantr 452 . . . . . 6  |-  ( (
ph  /\  N  = 
0 )  ->  A  e.  Fin )
1716olcd 383 . . . . 5  |-  ( (
ph  /\  N  = 
0 )  ->  ( A  C_  ( ZZ>= `  0
)  \/  A  e. 
Fin ) )
18 sumz 12516 . . . . 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 2468 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  sum_ k  e.  A  B  = 
0 )
213, 4, 203brtr4d 4242 . 2  |-  ( (
ph  /\  N  = 
0 )  ->  N  || 
sum_ k  e.  A  B )
2215adantr 452 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  A  e.  Fin )
23 fsumdvds.2 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
2423adantr 452 . . . . . 6  |-  ( (
ph  /\  N  =/=  0 )  ->  N  e.  ZZ )
2524zcnd 10376 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  N  e.  CC )
269adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  B  e.  ZZ )
2726zcnd 10376 . . . . 5  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  B  e.  CC )
28 simpr 448 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  N  =/=  0 )
2922, 25, 27, 28fsumdivc 12569 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  ( sum_ k  e.  A  B  /  N )  =  sum_ k  e.  A  ( B  /  N ) )
306adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  N  ||  B )
3124adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  N  e.  ZZ )
32 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  N  =/=  0 )
33 dvdsval2 12855 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  B  e.  ZZ )  ->  ( N  ||  B  <->  ( B  /  N )  e.  ZZ ) )
3431, 32, 26, 33syl3anc 1184 . . . . . 6  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  ( N  ||  B  <->  ( B  /  N )  e.  ZZ ) )
3530, 34mpbid 202 . . . . 5  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  ( B  /  N )  e.  ZZ )
3622, 35fsumzcl 12529 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  sum_ k  e.  A  ( B  /  N )  e.  ZZ )
3729, 36eqeltrd 2510 . . 3  |-  ( (
ph  /\  N  =/=  0 )  ->  ( sum_ k  e.  A  B  /  N )  e.  ZZ )
3815, 9fsumzcl 12529 . . . . 5  |-  ( ph  -> 
sum_ k  e.  A  B  e.  ZZ )
3938adantr 452 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  sum_ k  e.  A  B  e.  ZZ )
40 dvdsval2 12855 . . . 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 1184 . . 3  |-  ( (
ph  /\  N  =/=  0 )  ->  ( N  ||  sum_ k  e.  A  B 
<->  ( sum_ k  e.  A  B  /  N )  e.  ZZ ) )
4237, 41mpbird 224 . 2  |-  ( (
ph  /\  N  =/=  0 )  ->  N  || 
sum_ k  e.  A  B )
4321, 42pm2.61dane 2682 1  |-  ( ph  ->  N  ||  sum_ k  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    C_ wss 3320   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Fincfn 7109   0cc0 8990    / cdiv 9677   ZZcz 10282   ZZ>=cuz 10488   sum_csu 12479    || cdivides 12852
This theorem is referenced by:  3dvds  12912  sylow1lem3  15234  sylow2alem2  15252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-dvds 12853
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