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Theorem arisum 13682
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 10818 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 1zzd 10916 . . . . . 6  |-  ( N  e.  NN  ->  1  e.  ZZ )
3 nnz 10907 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  ZZ )
4 elfzelz 11713 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  ZZ )
54zcnd 10991 . . . . . . 7  |-  ( k  e.  ( 1 ... N )  ->  k  e.  CC )
65adantl 466 . . . . . 6  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... N ) )  ->  k  e.  CC )
7 id 22 . . . . . 6  |-  ( k  =  ( j  +  1 )  ->  k  =  ( j  +  1 ) )
82, 2, 3, 6, 7fsumshftm 13607 . . . . 5  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ j  e.  ( (
1  -  1 ) ... ( N  - 
1 ) ) ( j  +  1 ) )
9 1m1e0 10625 . . . . . . 7  |-  ( 1  -  1 )  =  0
109oveq1i 6306 . . . . . 6  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
1110sumeq1i 13531 . . . . 5  |-  sum_ j  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ( j  +  1 )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( j  +  1 )
128, 11syl6eq 2514 . . . 4  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( j  +  1 ) )
13 elfznn0 11796 . . . . . . . . 9  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  j  e.  NN0 )
1413adantl 466 . . . . . . . 8  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  j  e.  NN0 )
15 bcnp1n 12394 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( ( j  +  1 )  _C  j )  =  ( j  +  1 ) )
1614, 15syl 16 . . . . . . 7  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( ( j  +  1 )  _C  j )  =  ( j  +  1 ) )
1714nn0cnd 10875 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  j  e.  CC )
18 ax-1cn 9567 . . . . . . . . 9  |-  1  e.  CC
19 addcom 9783 . . . . . . . . 9  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( j  +  1 )  =  ( 1  +  j ) )
2017, 18, 19sylancl 662 . . . . . . . 8  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( j  +  1 )  =  ( 1  +  j ) )
2120oveq1d 6311 . . . . . . 7  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( ( j  +  1 )  _C  j )  =  ( ( 1  +  j )  _C  j ) )
2216, 21eqtr3d 2500 . . . . . 6  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( j  +  1 )  =  ( ( 1  +  j )  _C  j ) )
2322sumeq2dv 13536 . . . . 5  |-  ( N  e.  NN  ->  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( j  +  1 )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( ( 1  +  j )  _C  j ) )
24 1nn0 10832 . . . . . 6  |-  1  e.  NN0
25 nnm1nn0 10858 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
26 bcxmas 13658 . . . . . 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 663 . . . . 5  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( ( 1  +  j )  _C  j
) )
2823, 27eqtr4d 2501 . . . 4  |-  ( N  e.  NN  ->  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( j  +  1 )  =  ( ( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) ) )
29 1cnd 9629 . . . . . . . 8  |-  ( N  e.  NN  ->  1  e.  CC )
30 nncn 10564 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  CC )
3129, 29, 30ppncand 9990 . . . . . . 7  |-  ( N  e.  NN  ->  (
( 1  +  1 )  +  ( N  -  1 ) )  =  ( 1  +  N ) )
32 addcom 9783 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  N  e.  CC )  ->  ( 1  +  N
)  =  ( N  +  1 ) )
3318, 30, 32sylancr 663 . . . . . . 7  |-  ( N  e.  NN  ->  (
1  +  N )  =  ( N  + 
1 ) )
3431, 33eqtrd 2498 . . . . . 6  |-  ( N  e.  NN  ->  (
( 1  +  1 )  +  ( N  -  1 ) )  =  ( N  + 
1 ) )
3534oveq1d 6311 . . . . 5  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  ( ( N  +  1 )  _C  ( N  -  1 ) ) )
36 nnnn0 10823 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
37 bcp1m1 12400 . . . . . 6  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  _C  ( N  - 
1 ) )  =  ( ( ( N  +  1 )  x.  N )  /  2
) )
3836, 37syl 16 . . . . 5  |-  ( N  e.  NN  ->  (
( N  +  1 )  _C  ( N  -  1 ) )  =  ( ( ( N  +  1 )  x.  N )  / 
2 ) )
3930, 29, 30adddird 9638 . . . . . . 7  |-  ( N  e.  NN  ->  (
( N  +  1 )  x.  N )  =  ( ( N  x.  N )  +  ( 1  x.  N
) ) )
40 sqval 12229 . . . . . . . . . 10  |-  ( N  e.  CC  ->  ( N ^ 2 )  =  ( N  x.  N
) )
4140eqcomd 2465 . . . . . . . . 9  |-  ( N  e.  CC  ->  ( N  x.  N )  =  ( N ^
2 ) )
42 mulid2 9611 . . . . . . . . 9  |-  ( N  e.  CC  ->  (
1  x.  N )  =  N )
4341, 42oveq12d 6314 . . . . . . . 8  |-  ( N  e.  CC  ->  (
( N  x.  N
)  +  ( 1  x.  N ) )  =  ( ( N ^ 2 )  +  N ) )
4430, 43syl 16 . . . . . . 7  |-  ( N  e.  NN  ->  (
( N  x.  N
)  +  ( 1  x.  N ) )  =  ( ( N ^ 2 )  +  N ) )
4539, 44eqtrd 2498 . . . . . 6  |-  ( N  e.  NN  ->  (
( N  +  1 )  x.  N )  =  ( ( N ^ 2 )  +  N ) )
4645oveq1d 6311 . . . . 5  |-  ( N  e.  NN  ->  (
( ( N  + 
1 )  x.  N
)  /  2 )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
4735, 38, 463eqtrd 2502 . . . 4  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
4812, 28, 473eqtrd 2502 . . 3  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
49 oveq2 6304 . . . . . . 7  |-  ( N  =  0  ->  (
1 ... N )  =  ( 1 ... 0
) )
50 fz10 11731 . . . . . . 7  |-  ( 1 ... 0 )  =  (/)
5149, 50syl6eq 2514 . . . . . 6  |-  ( N  =  0  ->  (
1 ... N )  =  (/) )
5251sumeq1d 13534 . . . . 5  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ k  e.  (/)  k )
53 sum0 13554 . . . . 5  |-  sum_ k  e.  (/)  k  =  0
5452, 53syl6eq 2514 . . . 4  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  0 )
55 sq0i 12262 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
56 id 22 . . . . . . . 8  |-  ( N  =  0  ->  N  =  0 )
5755, 56oveq12d 6314 . . . . . . 7  |-  ( N  =  0  ->  (
( N ^ 2 )  +  N )  =  ( 0  +  0 ) )
58 00id 9772 . . . . . . 7  |-  ( 0  +  0 )  =  0
5957, 58syl6eq 2514 . . . . . 6  |-  ( N  =  0  ->  (
( N ^ 2 )  +  N )  =  0 )
6059oveq1d 6311 . . . . 5  |-  ( N  =  0  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  ( 0  / 
2 ) )
61 2cn 10627 . . . . . 6  |-  2  e.  CC
62 2ne0 10649 . . . . . 6  |-  2  =/=  0
6361, 62div0i 10299 . . . . 5  |-  ( 0  /  2 )  =  0
6460, 63syl6eq 2514 . . . 4  |-  ( N  =  0  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  0 )
6554, 64eqtr4d 2501 . . 3  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
6648, 65jaoi 379 . 2  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  sum_ k  e.  ( 1 ... N ) k  =  ( ( ( N ^ 2 )  +  N )  /  2 ) )
671, 66sylbi 195 1  |-  ( N  e.  NN0  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   (/)c0 3793  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    - cmin 9824    / cdiv 10227   NNcn 10556   2c2 10606   NN0cn0 10816   ...cfz 11697   ^cexp 12168    _C cbc 12382   sum_csu 13519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11821  df-seq 12110  df-exp 12169  df-fac 12356  df-bc 12383  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520
This theorem is referenced by:  arisum2  13683
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