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Theorem arisum 11508
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 9180 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 1zzd 9282 . . . . . 6  |-  ( N  e.  NN  ->  1  e.  ZZ )
3 nnz 9274 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  ZZ )
4 elfzelz 10027 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  ZZ )
54zcnd 9378 . . . . . . 7  |-  ( k  e.  ( 1 ... N )  ->  k  e.  CC )
65adantl 277 . . . . . 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 11455 . . . . 5  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ j  e.  ( (
1  -  1 ) ... ( N  - 
1 ) ) ( j  +  1 ) )
9 1m1e0 8990 . . . . . . 7  |-  ( 1  -  1 )  =  0
109oveq1i 5887 . . . . . 6  |-  ( ( 1  -  1 ) ... ( N  - 
1 ) )  =  ( 0 ... ( N  -  1 ) )
1110sumeq1i 11373 . . . . 5  |-  sum_ j  e.  ( ( 1  -  1 ) ... ( N  -  1 ) ) ( j  +  1 )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( j  +  1 )
128, 11eqtrdi 2226 . . . 4  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( j  +  1 ) )
13 elfznn0 10116 . . . . . . . . 9  |-  ( j  e.  ( 0 ... ( N  -  1 ) )  ->  j  e.  NN0 )
1413adantl 277 . . . . . . . 8  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  j  e.  NN0 )
15 bcnp1n 10741 . . . . . . . 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 9233 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  j  e.  CC )
18 ax-1cn 7906 . . . . . . . . 9  |-  1  e.  CC
19 addcom 8096 . . . . . . . . 9  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( j  +  1 )  =  ( 1  +  j ) )
2017, 18, 19sylancl 413 . . . . . . . 8  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( j  +  1 )  =  ( 1  +  j ) )
2120oveq1d 5892 . . . . . . 7  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( ( j  +  1 )  _C  j )  =  ( ( 1  +  j )  _C  j ) )
2216, 21eqtr3d 2212 . . . . . 6  |-  ( ( N  e.  NN  /\  j  e.  ( 0 ... ( N  - 
1 ) ) )  ->  ( j  +  1 )  =  ( ( 1  +  j )  _C  j ) )
2322sumeq2dv 11378 . . . . 5  |-  ( N  e.  NN  ->  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( j  +  1 )  =  sum_ j  e.  ( 0 ... ( N  - 
1 ) ) ( ( 1  +  j )  _C  j ) )
24 1nn0 9194 . . . . . 6  |-  1  e.  NN0
25 nnm1nn0 9219 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
26 bcxmas 11499 . . . . . 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 414 . . . . 5  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( ( 1  +  j )  _C  j
) )
2823, 27eqtr4d 2213 . . . 4  |-  ( N  e.  NN  ->  sum_ j  e.  ( 0 ... ( N  -  1 ) ) ( j  +  1 )  =  ( ( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) ) )
29 1cnd 7975 . . . . . . 7  |-  ( N  e.  NN  ->  1  e.  CC )
30 nncn 8929 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  CC )
3129, 29, 30ppncand 8310 . . . . . . 7  |-  ( N  e.  NN  ->  (
( 1  +  1 )  +  ( N  -  1 ) )  =  ( 1  +  N ) )
3229, 30, 31comraddd 8116 . . . . . 6  |-  ( N  e.  NN  ->  (
( 1  +  1 )  +  ( N  -  1 ) )  =  ( N  + 
1 ) )
3332oveq1d 5892 . . . . 5  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  ( ( N  +  1 )  _C  ( N  -  1 ) ) )
34 nnnn0 9185 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
35 bcp1m1 10747 . . . . . 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 10580 . . . . . . . . . 10  |-  ( N  e.  CC  ->  ( N ^ 2 )  =  ( N  x.  N
) )
3837eqcomd 2183 . . . . . . . . 9  |-  ( N  e.  CC  ->  ( N  x.  N )  =  ( N ^
2 ) )
39 mullid 7957 . . . . . . . . 9  |-  ( N  e.  CC  ->  (
1  x.  N )  =  N )
4038, 39oveq12d 5895 . . . . . . . 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 7987 . . . . . 6  |-  ( N  e.  NN  ->  (
( N  +  1 )  x.  N )  =  ( ( N ^ 2 )  +  N ) )
4342oveq1d 5892 . . . . 5  |-  ( N  e.  NN  ->  (
( ( N  + 
1 )  x.  N
)  /  2 )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
4433, 36, 433eqtrd 2214 . . . 4  |-  ( N  e.  NN  ->  (
( ( 1  +  1 )  +  ( N  -  1 ) )  _C  ( N  -  1 ) )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
4512, 28, 443eqtrd 2214 . . 3  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
46 oveq2 5885 . . . . . . 7  |-  ( N  =  0  ->  (
1 ... N )  =  ( 1 ... 0
) )
47 fz10 10048 . . . . . . 7  |-  ( 1 ... 0 )  =  (/)
4846, 47eqtrdi 2226 . . . . . 6  |-  ( N  =  0  ->  (
1 ... N )  =  (/) )
4948sumeq1d 11376 . . . . 5  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  sum_ k  e.  (/)  k )
50 sum0 11398 . . . . 5  |-  sum_ k  e.  (/)  k  =  0
5149, 50eqtrdi 2226 . . . 4  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  0 )
52 sq0i 10614 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
53 id 19 . . . . . . . 8  |-  ( N  =  0  ->  N  =  0 )
5452, 53oveq12d 5895 . . . . . . 7  |-  ( N  =  0  ->  (
( N ^ 2 )  +  N )  =  ( 0  +  0 ) )
55 00id 8100 . . . . . . 7  |-  ( 0  +  0 )  =  0
5654, 55eqtrdi 2226 . . . . . 6  |-  ( N  =  0  ->  (
( N ^ 2 )  +  N )  =  0 )
5756oveq1d 5892 . . . . 5  |-  ( N  =  0  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  ( 0  / 
2 ) )
58 2cn 8992 . . . . . 6  |-  2  e.  CC
59 2ap0 9014 . . . . . 6  |-  2 #  0
6058, 59div0api 8705 . . . . 5  |-  ( 0  /  2 )  =  0
6157, 60eqtrdi 2226 . . . 4  |-  ( N  =  0  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  0 )
6251, 61eqtr4d 2213 . . 3  |-  ( N  =  0  ->  sum_ k  e.  ( 1 ... N
) k  =  ( ( ( N ^
2 )  +  N
)  /  2 ) )
6345, 62jaoi 716 . 2  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  sum_ k  e.  ( 1 ... N ) k  =  ( ( ( N ^ 2 )  +  N )  /  2 ) )
641, 63sylbi 121 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 104    \/ wo 708    = wceq 1353    e. wcel 2148   (/)c0 3424  (class class class)co 5877   CCcc 7811   0cc0 7813   1c1 7814    + caddc 7816    x. cmul 7818    - cmin 8130    / cdiv 8631   NNcn 8921   2c2 8972   NN0cn0 9178   ...cfz 10010   ^cexp 10521    _C cbc 10729   sum_csu 11363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-seqfrec 10448  df-exp 10522  df-fac 10708  df-bc 10730  df-ihash 10758  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-sumdc 11364
This theorem is referenced by:  arisum2  11509
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