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Theorem arisum 12320
Description: Arithmetic series sum of the first  N positive integers. (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 9969 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 1z 10055 . . . . . . 7  |-  1  e.  ZZ
32a1i 10 . . . . . 6  |-  ( N  e.  NN  ->  1  e.  ZZ )
4 nnz 10047 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  ZZ )
5 elfzelz 10800 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  ZZ )
65zcnd 10120 . . . . . . 7  |-  ( k  e.  ( 1 ... N )  ->  k  e.  CC )
76adantl 452 . . . . . 6  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  ->  k  e.  CC )
8 id 19 . . . . . 6  |-  ( k  =  ( j  +  1 )  ->  k  =  ( j  +  1 ) )
93, 3, 4, 7, 8fsumshftm 12245 . . . . 5  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ j  e.  ( (
1  -  1 ) ... ( N  - 
1 ) ) ( j  +  1 ) )
10 1m1e0 9816 . . . . . . 7  |-  ( 1  -  1 )  =  0
1110oveq1i 5870 . . . . . 6  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
1211sumeq1i 12173 . . . . 5  |-  sum_ j  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ( j  +  1 )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( j  +  1 )
139, 12syl6eq 2333 . . . 4  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( j  +  1 ) )
14 elfznn0 10824 . . . . . . . . 9  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  j  e.  NN0 )
1514adantl 452 . . . . . . . 8  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  j  e.  NN0 )
16 bcnp1n 11328 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( ( j  +  1 )  _C  j )  =  ( j  +  1 ) )
1715, 16syl 15 . . . . . . 7  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( ( j  +  1 )  _C  j )  =  ( j  +  1 ) )
1815nn0cnd 10022 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  j  e.  CC )
19 ax-1cn 8797 . . . . . . . . 9  |-  1  e.  CC
20 addcom 9000 . . . . . . . . 9  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( j  +  1 )  =  ( 1  +  j ) )
2118, 19, 20sylancl 643 . . . . . . . 8  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( j  +  1 )  =  ( 1  +  j ) )
2221oveq1d 5875 . . . . . . 7  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( ( j  +  1 )  _C  j )  =  ( ( 1  +  j )  _C  j ) )
2317, 22eqtr3d 2319 . . . . . 6  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( j  +  1 )  =  ( ( 1  +  j )  _C  j ) )
2423sumeq2dv 12178 . . . . 5  |-  ( N  e.  NN  ->  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( j  +  1 )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( ( 1  +  j )  _C  j ) )
25 1nn0 9983 . . . . . 6  |-  1  e.  NN0
26 nnm1nn0 10007 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
27 bcxmas 12296 . . . . . 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 ) )
2825, 26, 27sylancr 644 . . . . 5  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( ( 1  +  j )  _C  j
) )
2924, 28eqtr4d 2320 . . . 4  |-  ( N  e.  NN  ->  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( j  +  1 )  =  ( ( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) ) )
3019a1i 10 . . . . . . . 8  |-  ( N  e.  NN  ->  1  e.  CC )
31 nncn 9756 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  CC )
3230, 30, 31ppncand 9199 . . . . . . 7  |-  ( N  e.  NN  ->  (
( 1  +  1 )  +  ( N  -  1 ) )  =  ( 1  +  N ) )
33 addcom 9000 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  N  e.  CC )  ->  ( 1  +  N
)  =  ( N  +  1 ) )
3419, 31, 33sylancr 644 . . . . . . 7  |-  ( N  e.  NN  ->  (
1  +  N )  =  ( N  + 
1 ) )
3532, 34eqtrd 2317 . . . . . 6  |-  ( N  e.  NN  ->  (
( 1  +  1 )  +  ( N  -  1 ) )  =  ( N  + 
1 ) )
3635oveq1d 5875 . . . . 5  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  ( ( N  +  1 )  _C  ( N  -  1 ) ) )
37 nnnn0 9974 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
38 bcp1m1 11334 . . . . . 6  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  _C  ( N  - 
1 ) )  =  ( ( ( N  +  1 )  x.  N )  /  2
) )
3937, 38syl 15 . . . . 5  |-  ( N  e.  NN  ->  (
( N  +  1 )  _C  ( N  -  1 ) )  =  ( ( ( N  +  1 )  x.  N )  / 
2 ) )
4031, 30, 31adddird 8862 . . . . . . 7  |-  ( N  e.  NN  ->  (
( N  +  1 )  x.  N )  =  ( ( N  x.  N )  +  ( 1  x.  N
) ) )
41 sqval 11165 . . . . . . . . . 10  |-  ( N  e.  CC  ->  ( N ^ 2 )  =  ( N  x.  N
) )
4241eqcomd 2290 . . . . . . . . 9  |-  ( N  e.  CC  ->  ( N  x.  N )  =  ( N ^
2 ) )
43 mulid2 8838 . . . . . . . . 9  |-  ( N  e.  CC  ->  (
1  x.  N )  =  N )
4442, 43oveq12d 5878 . . . . . . . 8  |-  ( N  e.  CC  ->  (
( N  x.  N
)  +  ( 1  x.  N ) )  =  ( ( N ^ 2 )  +  N ) )
4531, 44syl 15 . . . . . . 7  |-  ( N  e.  NN  ->  (
( N  x.  N
)  +  ( 1  x.  N ) )  =  ( ( N ^ 2 )  +  N ) )
4640, 45eqtrd 2317 . . . . . 6  |-  ( N  e.  NN  ->  (
( N  +  1 )  x.  N )  =  ( ( N ^ 2 )  +  N ) )
4746oveq1d 5875 . . . . 5  |-  ( N  e.  NN  ->  (
( ( N  + 
1 )  x.  N
)  /  2 )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
4836, 39, 473eqtrd 2321 . . . 4  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
4913, 29, 483eqtrd 2321 . . 3  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
50 oveq2 5868 . . . . . . 7  |-  ( N  =  0  ->  (
1 ... N )  =  ( 1 ... 0
) )
51 fz10 10816 . . . . . . 7  |-  ( 1 ... 0 )  =  (/)
5250, 51syl6eq 2333 . . . . . 6  |-  ( N  =  0  ->  (
1 ... N )  =  (/) )
5352sumeq1d 12176 . . . . 5  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ k  e.  (/)  k )
54 sum0 12196 . . . . 5  |-  sum_ k  e.  (/)  k  =  0
5553, 54syl6eq 2333 . . . 4  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  0 )
56 sq0i 11198 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
57 id 19 . . . . . . . 8  |-  ( N  =  0  ->  N  =  0 )
5856, 57oveq12d 5878 . . . . . . 7  |-  ( N  =  0  ->  (
( N ^ 2 )  +  N )  =  ( 0  +  0 ) )
59 00id 8989 . . . . . . 7  |-  ( 0  +  0 )  =  0
6058, 59syl6eq 2333 . . . . . 6  |-  ( N  =  0  ->  (
( N ^ 2 )  +  N )  =  0 )
6160oveq1d 5875 . . . . 5  |-  ( N  =  0  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  ( 0  / 
2 ) )
62 2cn 9818 . . . . . 6  |-  2  e.  CC
63 2ne0 9831 . . . . . 6  |-  2  =/=  0
6462, 63div0i 9496 . . . . 5  |-  ( 0  /  2 )  =  0
6561, 64syl6eq 2333 . . . 4  |-  ( N  =  0  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  0 )
6655, 65eqtr4d 2320 . . 3  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
6749, 66jaoi 368 . 2  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  sum_ k  e.  ( 1 ... N ) k  =  ( ( ( N ^ 2 )  +  N )  /  2 ) )
681, 67sylbi 187 1  |-  ( N  e.  NN0  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1625    e. wcel 1686   (/)c0 3457  (class class class)co 5860   CCcc 8737   0cc0 8739   1c1 8740    + caddc 8742    x. cmul 8744    - cmin 9039    / cdiv 9425   NNcn 9748   2c2 9797   NN0cn0 9967   ZZcz 10026   ...cfz 10784   ^cexp 11106    _C cbc 11317   sum_csu 12160
This theorem is referenced by:  arisum2  12321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-fz 10785  df-fzo 10873  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-sum 12161
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