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Theorem fsumdvds 12572
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 10035 . . . 4  |-  0  e.  ZZ
2 dvds0 12544 . . . 4  |-  ( 0  e.  ZZ  ->  0  ||  0 )
31, 2mp1i 11 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  0  ||  0 )
4 simpr 447 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  N  =  0 )
5 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  N  =  0 )
6 fsumdvds.4 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  N  ||  B )
76adantlr 695 . . . . . . 7  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  N  ||  B )
85, 7eqbrtrrd 4045 . . . . . 6  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  0  ||  B )
9 fsumdvds.3 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ZZ )
109adantlr 695 . . . . . . 7  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  B  e.  ZZ )
11 0dvds 12549 . . . . . . 7  |-  ( B  e.  ZZ  ->  (
0  ||  B  <->  B  = 
0 ) )
1210, 11syl 15 . . . . . 6  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  ( 0  ||  B  <->  B  =  0 ) )
138, 12mpbid 201 . . . . 5  |-  ( ( ( ph  /\  N  =  0 )  /\  k  e.  A )  ->  B  =  0 )
1413sumeq2dv 12176 . . . 4  |-  ( (
ph  /\  N  = 
0 )  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  0 )
15 fsumdvds.1 . . . . . . 7  |-  ( ph  ->  A  e.  Fin )
1615adantr 451 . . . . . 6  |-  ( (
ph  /\  N  = 
0 )  ->  A  e.  Fin )
1716olcd 382 . . . . 5  |-  ( (
ph  /\  N  = 
0 )  ->  ( A  C_  ( ZZ>= `  0
)  \/  A  e. 
Fin ) )
18 sumz 12195 . . . . 5  |-  ( ( A  C_  ( ZZ>= ` 
0 )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  = 
0 )
1917, 18syl 15 . . . 4  |-  ( (
ph  /\  N  = 
0 )  ->  sum_ k  e.  A  0  = 
0 )
2014, 19eqtrd 2315 . . 3  |-  ( (
ph  /\  N  = 
0 )  ->  sum_ k  e.  A  B  = 
0 )
213, 4, 203brtr4d 4053 . 2  |-  ( (
ph  /\  N  = 
0 )  ->  N  || 
sum_ k  e.  A  B )
2215adantr 451 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  A  e.  Fin )
23 fsumdvds.2 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
2423adantr 451 . . . . . 6  |-  ( (
ph  /\  N  =/=  0 )  ->  N  e.  ZZ )
2524zcnd 10118 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  N  e.  CC )
269adantlr 695 . . . . . 6  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  B  e.  ZZ )
2726zcnd 10118 . . . . 5  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  B  e.  CC )
28 simpr 447 . . . . 5  |-  ( (
ph  /\  N  =/=  0 )  ->  N  =/=  0 )
2922, 25, 27, 28fsumdivc 12248 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  ( sum_ k  e.  A  B  /  N )  =  sum_ k  e.  A  ( B  /  N ) )
306adantlr 695 . . . . . 6  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  N  ||  B )
3124adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  N  e.  ZZ )
32 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  N  =/=  0 )
33 dvdsval2 12534 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  B  e.  ZZ )  ->  ( N  ||  B  <->  ( B  /  N )  e.  ZZ ) )
3431, 32, 26, 33syl3anc 1182 . . . . . 6  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  ( N  ||  B  <->  ( B  /  N )  e.  ZZ ) )
3530, 34mpbid 201 . . . . 5  |-  ( ( ( ph  /\  N  =/=  0 )  /\  k  e.  A )  ->  ( B  /  N )  e.  ZZ )
3622, 35fsumzcl 12208 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  sum_ k  e.  A  ( B  /  N )  e.  ZZ )
3729, 36eqeltrd 2357 . . 3  |-  ( (
ph  /\  N  =/=  0 )  ->  ( sum_ k  e.  A  B  /  N )  e.  ZZ )
3815, 9fsumzcl 12208 . . . . 5  |-  ( ph  -> 
sum_ k  e.  A  B  e.  ZZ )
3938adantr 451 . . . 4  |-  ( (
ph  /\  N  =/=  0 )  ->  sum_ k  e.  A  B  e.  ZZ )
40 dvdsval2 12534 . . . 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 1182 . . 3  |-  ( (
ph  /\  N  =/=  0 )  ->  ( N  ||  sum_ k  e.  A  B 
<->  ( sum_ k  e.  A  B  /  N )  e.  ZZ ) )
4237, 41mpbird 223 . 2  |-  ( (
ph  /\  N  =/=  0 )  ->  N  || 
sum_ k  e.  A  B )
4321, 42pm2.61dane 2524 1  |-  ( ph  ->  N  ||  sum_ k  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Fincfn 6863   0cc0 8737    / cdiv 9423   ZZcz 10024   ZZ>=cuz 10230   sum_csu 12158    || cdivides 12531
This theorem is referenced by:  3dvds  12591  sylow1lem3  14911  sylow2alem2  14929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532
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