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Theorem arisum 13320
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 10579 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 1zzd 10675 . . . . . 6  |-  ( N  e.  NN  ->  1  e.  ZZ )
3 nnz 10666 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  ZZ )
4 elfzelz 11451 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  ZZ )
54zcnd 10746 . . . . . . 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 13246 . . . . 5  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ j  e.  ( (
1  -  1 ) ... ( N  - 
1 ) ) ( j  +  1 ) )
9 1m1e0 10388 . . . . . . 7  |-  ( 1  -  1 )  =  0
109oveq1i 6099 . . . . . 6  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
1110sumeq1i 13173 . . . . 5  |-  sum_ j  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ( j  +  1 )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( j  +  1 )
128, 11syl6eq 2489 . . . 4  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( j  +  1 ) )
13 elfznn0 11479 . . . . . . . . 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 12088 . . . . . . . 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 10636 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  j  e.  CC )
18 ax-1cn 9338 . . . . . . . . 9  |-  1  e.  CC
19 addcom 9553 . . . . . . . . 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 6104 . . . . . . 7  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( ( j  +  1 )  _C  j )  =  ( ( 1  +  j )  _C  j ) )
2216, 21eqtr3d 2475 . . . . . 6  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( j  +  1 )  =  ( ( 1  +  j )  _C  j ) )
2322sumeq2dv 13178 . . . . 5  |-  ( N  e.  NN  ->  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( j  +  1 )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( ( 1  +  j )  _C  j ) )
24 1nn0 10593 . . . . . 6  |-  1  e.  NN0
25 nnm1nn0 10619 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
26 bcxmas 13296 . . . . . 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 2476 . . . 4  |-  ( N  e.  NN  ->  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( j  +  1 )  =  ( ( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) ) )
29 1cnd 9400 . . . . . . . 8  |-  ( N  e.  NN  ->  1  e.  CC )
30 nncn 10328 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  CC )
3129, 29, 30ppncand 9757 . . . . . . 7  |-  ( N  e.  NN  ->  (
( 1  +  1 )  +  ( N  -  1 ) )  =  ( 1  +  N ) )
32 addcom 9553 . . . . . . . 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 2473 . . . . . 6  |-  ( N  e.  NN  ->  (
( 1  +  1 )  +  ( N  -  1 ) )  =  ( N  + 
1 ) )
3534oveq1d 6104 . . . . 5  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  ( ( N  +  1 )  _C  ( N  -  1 ) ) )
36 nnnn0 10584 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
37 bcp1m1 12094 . . . . . 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 9409 . . . . . . 7  |-  ( N  e.  NN  ->  (
( N  +  1 )  x.  N )  =  ( ( N  x.  N )  +  ( 1  x.  N
) ) )
40 sqval 11923 . . . . . . . . . 10  |-  ( N  e.  CC  ->  ( N ^ 2 )  =  ( N  x.  N
) )
4140eqcomd 2446 . . . . . . . . 9  |-  ( N  e.  CC  ->  ( N  x.  N )  =  ( N ^
2 ) )
42 mulid2 9382 . . . . . . . . 9  |-  ( N  e.  CC  ->  (
1  x.  N )  =  N )
4341, 42oveq12d 6107 . . . . . . . 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 2473 . . . . . 6  |-  ( N  e.  NN  ->  (
( N  +  1 )  x.  N )  =  ( ( N ^ 2 )  +  N ) )
4645oveq1d 6104 . . . . 5  |-  ( N  e.  NN  ->  (
( ( N  + 
1 )  x.  N
)  /  2 )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
4735, 38, 463eqtrd 2477 . . . 4  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
4812, 28, 473eqtrd 2477 . . 3  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
49 oveq2 6097 . . . . . . 7  |-  ( N  =  0  ->  (
1 ... N )  =  ( 1 ... 0
) )
50 fz10 11468 . . . . . . 7  |-  ( 1 ... 0 )  =  (/)
5149, 50syl6eq 2489 . . . . . 6  |-  ( N  =  0  ->  (
1 ... N )  =  (/) )
5251sumeq1d 13176 . . . . 5  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ k  e.  (/)  k )
53 sum0 13196 . . . . 5  |-  sum_ k  e.  (/)  k  =  0
5452, 53syl6eq 2489 . . . 4  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  0 )
55 sq0i 11956 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
56 id 22 . . . . . . . 8  |-  ( N  =  0  ->  N  =  0 )
5755, 56oveq12d 6107 . . . . . . 7  |-  ( N  =  0  ->  (
( N ^ 2 )  +  N )  =  ( 0  +  0 ) )
58 00id 9542 . . . . . . 7  |-  ( 0  +  0 )  =  0
5957, 58syl6eq 2489 . . . . . 6  |-  ( N  =  0  ->  (
( N ^ 2 )  +  N )  =  0 )
6059oveq1d 6104 . . . . 5  |-  ( N  =  0  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  ( 0  / 
2 ) )
61 2cn 10390 . . . . . 6  |-  2  e.  CC
62 2ne0 10412 . . . . . 6  |-  2  =/=  0
6361, 62div0i 10063 . . . . 5  |-  ( 0  /  2 )  =  0
6460, 63syl6eq 2489 . . . 4  |-  ( N  =  0  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  0 )
6554, 64eqtr4d 2476 . . 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 1369    e. wcel 1756   (/)c0 3635  (class class class)co 6089   CCcc 9278   0cc0 9280   1c1 9281    + caddc 9283    x. cmul 9285    - cmin 9593    / cdiv 9991   NNcn 10320   2c2 10369   NN0cn0 10577   ...cfz 11435   ^cexp 11863    _C cbc 12076   sum_csu 13161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-oi 7722  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-fz 11436  df-fzo 11547  df-seq 11805  df-exp 11864  df-fac 12050  df-bc 12077  df-hash 12102  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-clim 12964  df-sum 13162
This theorem is referenced by:  arisum2  13321
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