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

Theorem fsumdvds 12848
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 10249 . . . 4  |-  0  e.  ZZ
2 dvds0 12820 . . . 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 4194 . . . . . 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 12825 . . . . . . 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 12452 . . . 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 12471 . . . . 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 2436 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  sum_ k  e.  A  B  = 
0 )
213, 4, 203brtr4d 4202 . 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 10332 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  N  e.  CC )
269adantlr 696 . . . . . 6  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  B  e.  ZZ )
2726zcnd 10332 . . . . 5  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  B  e.  CC )
28 simpr 448 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  N  =/=  0 )
2922, 25, 27, 28fsumdivc 12524 . . . 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 12810 . . . . . . 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 12484 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  sum_ k  e.  A  ( B  /  N )  e.  ZZ )
3729, 36eqeltrd 2478 . . 3  |-  ( (
ph  /\  N  =/=  0 )  ->  ( sum_ k  e.  A  B  /  N )  e.  ZZ )
3815, 9fsumzcl 12484 . . . . 5  |-  ( ph  -> 
sum_ k  e.  A  B  e.  ZZ )
3938adantr 452 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  sum_ k  e.  A  B  e.  ZZ )
40 dvdsval2 12810 . . . 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 2645 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 1649    e. wcel 1721    =/= wne 2567    C_ wss 3280   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Fincfn 7068   0cc0 8946    / cdiv 9633   ZZcz 10238   ZZ>=cuz 10444   sum_csu 12434    || cdivides 12807
This theorem is referenced by:  3dvds  12867  sylow1lem3  15189  sylow2alem2  15207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-dvds 12808
  Copyright terms: Public domain W3C validator