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Theorem arisum 10892
Description: Arithmetic series sum of the first  N positive integers. This is Metamath 100 proof #68. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 22-May-2014.)
Assertion
Ref Expression
arisum  |-  ( N  e.  NN0  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
Distinct variable group:    k, N

Proof of Theorem arisum
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 elnn0 8675 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 1zzd 8777 . . . . . 6  |-  ( N  e.  NN  ->  1  e.  ZZ )
3 nnz 8769 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  ZZ )
4 elfzelz 9440 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  ZZ )
54zcnd 8869 . . . . . . 7  |-  ( k  e.  ( 1 ... N )  ->  k  e.  CC )
65adantl 271 . . . . . 6  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  ->  k  e.  CC )
7 id 19 . . . . . 6  |-  ( k  =  ( j  +  1 )  ->  k  =  ( j  +  1 ) )
82, 2, 3, 6, 7fsumshftm 10839 . . . . 5  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ j  e.  ( (
1  -  1 ) ... ( N  - 
1 ) ) ( j  +  1 ) )
9 1m1e0 8491 . . . . . . 7  |-  ( 1  -  1 )  =  0
109oveq1i 5662 . . . . . 6  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
1110sumeq1i 10752 . . . . 5  |-  sum_ j  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ( j  +  1 )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( j  +  1 )
128, 11syl6eq 2136 . . . 4  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( j  +  1 ) )
13 elfznn0 9528 . . . . . . . . 9  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  j  e.  NN0 )
1413adantl 271 . . . . . . . 8  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  j  e.  NN0 )
15 bcnp1n 10167 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( ( j  +  1 )  _C  j )  =  ( j  +  1 ) )
1614, 15syl 14 . . . . . . 7  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( ( j  +  1 )  _C  j )  =  ( j  +  1 ) )
1714nn0cnd 8728 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  j  e.  CC )
18 ax-1cn 7438 . . . . . . . . 9  |-  1  e.  CC
19 addcom 7619 . . . . . . . . 9  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( j  +  1 )  =  ( 1  +  j ) )
2017, 18, 19sylancl 404 . . . . . . . 8  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( j  +  1 )  =  ( 1  +  j ) )
2120oveq1d 5667 . . . . . . 7  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( ( j  +  1 )  _C  j )  =  ( ( 1  +  j )  _C  j ) )
2216, 21eqtr3d 2122 . . . . . 6  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( j  +  1 )  =  ( ( 1  +  j )  _C  j ) )
2322sumeq2dv 10757 . . . . 5  |-  ( N  e.  NN  ->  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( j  +  1 )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( ( 1  +  j )  _C  j ) )
24 1nn0 8689 . . . . . 6  |-  1  e.  NN0
25 nnm1nn0 8714 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
26 bcxmas 10883 . . . . . 6  |-  ( ( 1  e.  NN0  /\  ( N  -  1
)  e.  NN0 )  ->  ( ( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( ( 1  +  j )  _C  j ) )
2724, 25, 26sylancr 405 . . . . 5  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( ( 1  +  j )  _C  j
) )
2823, 27eqtr4d 2123 . . . 4  |-  ( N  e.  NN  ->  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( j  +  1 )  =  ( ( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) ) )
29 1cnd 7504 . . . . . . 7  |-  ( N  e.  NN  ->  1  e.  CC )
30 nncn 8430 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  CC )
3129, 29, 30ppncand 7833 . . . . . . 7  |-  ( N  e.  NN  ->  (
( 1  +  1 )  +  ( N  -  1 ) )  =  ( 1  +  N ) )
3229, 30, 31comraddd 7639 . . . . . 6  |-  ( N  e.  NN  ->  (
( 1  +  1 )  +  ( N  -  1 ) )  =  ( N  + 
1 ) )
3332oveq1d 5667 . . . . 5  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  ( ( N  +  1 )  _C  ( N  -  1 ) ) )
34 nnnn0 8680 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
35 bcp1m1 10173 . . . . . 6  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  _C  ( N  - 
1 ) )  =  ( ( ( N  +  1 )  x.  N )  /  2
) )
3634, 35syl 14 . . . . 5  |-  ( N  e.  NN  ->  (
( N  +  1 )  _C  ( N  -  1 ) )  =  ( ( ( N  +  1 )  x.  N )  / 
2 ) )
37 sqval 10013 . . . . . . . . . 10  |-  ( N  e.  CC  ->  ( N ^ 2 )  =  ( N  x.  N
) )
3837eqcomd 2093 . . . . . . . . 9  |-  ( N  e.  CC  ->  ( N  x.  N )  =  ( N ^
2 ) )
39 mulid2 7486 . . . . . . . . 9  |-  ( N  e.  CC  ->  (
1  x.  N )  =  N )
4038, 39oveq12d 5670 . . . . . . . 8  |-  ( N  e.  CC  ->  (
( N  x.  N
)  +  ( 1  x.  N ) )  =  ( ( N ^ 2 )  +  N ) )
4130, 40syl 14 . . . . . . 7  |-  ( N  e.  NN  ->  (
( N  x.  N
)  +  ( 1  x.  N ) )  =  ( ( N ^ 2 )  +  N ) )
4230, 30, 29, 41joinlmuladdmuld 7515 . . . . . 6  |-  ( N  e.  NN  ->  (
( N  +  1 )  x.  N )  =  ( ( N ^ 2 )  +  N ) )
4342oveq1d 5667 . . . . 5  |-  ( N  e.  NN  ->  (
( ( N  + 
1 )  x.  N
)  /  2 )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
4433, 36, 433eqtrd 2124 . . . 4  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
4512, 28, 443eqtrd 2124 . . 3  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
46 oveq2 5660 . . . . . . 7  |-  ( N  =  0  ->  (
1 ... N )  =  ( 1 ... 0
) )
47 fz10 9460 . . . . . . 7  |-  ( 1 ... 0 )  =  (/)
4846, 47syl6eq 2136 . . . . . 6  |-  ( N  =  0  ->  (
1 ... N )  =  (/) )
4948sumeq1d 10755 . . . . 5  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ k  e.  (/)  k )
50 sum0 10780 . . . . 5  |-  sum_ k  e.  (/)  k  =  0
5149, 50syl6eq 2136 . . . 4  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  0 )
52 sq0i 10046 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
53 id 19 . . . . . . . 8  |-  ( N  =  0  ->  N  =  0 )
5452, 53oveq12d 5670 . . . . . . 7  |-  ( N  =  0  ->  (
( N ^ 2 )  +  N )  =  ( 0  +  0 ) )
55 00id 7623 . . . . . . 7  |-  ( 0  +  0 )  =  0
5654, 55syl6eq 2136 . . . . . 6  |-  ( N  =  0  ->  (
( N ^ 2 )  +  N )  =  0 )
5756oveq1d 5667 . . . . 5  |-  ( N  =  0  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  ( 0  / 
2 ) )
58 2cn 8493 . . . . . 6  |-  2  e.  CC
59 2ap0 8515 . . . . . 6  |-  2 #  0
6058, 59div0api 8213 . . . . 5  |-  ( 0  /  2 )  =  0
6157, 60syl6eq 2136 . . . 4  |-  ( N  =  0  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  0 )
6251, 61eqtr4d 2123 . . 3  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
6345, 62jaoi 671 . 2  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  sum_ k  e.  ( 1 ... N ) k  =  ( ( ( N ^ 2 )  +  N )  /  2 ) )
641, 63sylbi 119 1  |-  ( N  e.  NN0  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438   (/)c0 3286  (class class class)co 5652   CCcc 7348   0cc0 7350   1c1 7351    + caddc 7353    x. cmul 7355    - cmin 7653    / cdiv 8139   NNcn 8422   2c2 8473   NN0cn0 8673   ...cfz 9424   ^cexp 9954    _C cbc 10155   sum_csu 10742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-exp 9955  df-fac 10134  df-bc 10156  df-ihash 10184  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667  df-isum 10743
This theorem is referenced by:  arisum2  10893
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