MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsumdvds Structured version   Unicode version

Theorem fsumdvds 12885
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 10285 . . . 4  |-  0  e.  ZZ
2 dvds0 12857 . . . 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 4226 . . . . . 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 12862 . . . . . . 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 12489 . . . 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 12508 . . . . 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 2467 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  sum_ k  e.  A  B  = 
0 )
213, 4, 203brtr4d 4234 . 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 10368 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  N  e.  CC )
269adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  B  e.  ZZ )
2726zcnd 10368 . . . . 5  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  B  e.  CC )
28 simpr 448 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  N  =/=  0 )
2922, 25, 27, 28fsumdivc 12561 . . . 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 12847 . . . . . . 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 12521 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  sum_ k  e.  A  ( B  /  N )  e.  ZZ )
3729, 36eqeltrd 2509 . . 3  |-  ( (
ph  /\  N  =/=  0 )  ->  ( sum_ k  e.  A  B  /  N )  e.  ZZ )
3815, 9fsumzcl 12521 . . . . 5  |-  ( ph  -> 
sum_ k  e.  A  B  e.  ZZ )
3938adantr 452 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  sum_ k  e.  A  B  e.  ZZ )
40 dvdsval2 12847 . . . 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 2676 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 2598    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Fincfn 7101   0cc0 8982    / cdiv 9669   ZZcz 10274   ZZ>=cuz 10480   sum_csu 12471    || cdivides 12844
This theorem is referenced by:  3dvds  12904  sylow1lem3  15226  sylow2alem2  15244
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-dvds 12845
  Copyright terms: Public domain W3C validator