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Theorem arisum2 13344
Description: Arithmetic series sum of the first  N nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
Assertion
Ref Expression
arisum2  |-  ( N  e.  NN0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
Distinct variable group:    k, N

Proof of Theorem arisum2
StepHypRef Expression
1 elnn0 10602 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnm1nn0 10642 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
3 nn0uz 10916 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
42, 3syl6eleq 2533 . . . . 5  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  ( ZZ>= `  0
) )
5 elfznn0 11502 . . . . . . 7  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
65adantl 466 . . . . . 6  |-  ( ( N  e.  NN  /\  k  e.  ( 0 ... ( N  - 
1 ) ) )  ->  k  e.  NN0 )
76nn0cnd 10659 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  ( 0 ... ( N  - 
1 ) ) )  ->  k  e.  CC )
8 id 22 . . . . 5  |-  ( k  =  0  ->  k  =  0 )
94, 7, 8fsum1p 13243 . . . 4  |-  ( N  e.  NN  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( 0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k ) )
10 1e0p1 10804 . . . . . . . . 9  |-  1  =  ( 0  +  1 )
1110oveq1i 6122 . . . . . . . 8  |-  ( 1 ... ( N  - 
1 ) )  =  ( ( 0  +  1 ) ... ( N  -  1 ) )
1211sumeq1i 13196 . . . . . . 7  |-  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  sum_ k  e.  ( (
0  +  1 ) ... ( N  - 
1 ) ) k
1312oveq2i 6123 . . . . . 6  |-  ( 0  +  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )  =  ( 0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )
14 fzfid 11816 . . . . . . . 8  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  e. 
Fin )
15 elfznn 11499 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( N  -  1 ) )  ->  k  e.  NN )
1615adantl 466 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... ( N  - 
1 ) ) )  ->  k  e.  NN )
1716nncnd 10359 . . . . . . . 8  |-  ( ( N  e.  NN  /\  k  e.  ( 1 ... ( N  - 
1 ) ) )  ->  k  e.  CC )
1814, 17fsumcl 13231 . . . . . . 7  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  e.  CC )
1918addid2d 9591 . . . . . 6  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )  = 
sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )
2013, 19syl5eqr 2489 . . . . 5  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )  = 
sum_ k  e.  ( 1 ... ( N  -  1 ) ) k )
21 arisum 13343 . . . . . . 7  |-  ( ( N  -  1 )  e.  NN0  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  ( ( ( ( N  -  1 ) ^
2 )  +  ( N  -  1 ) )  /  2 ) )
222, 21syl 16 . . . . . 6  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  ( ( ( ( N  -  1 ) ^
2 )  +  ( N  -  1 ) )  /  2 ) )
23 nncn 10351 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN  ->  N  e.  CC )
2423mulid1d 9424 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  ( N  x.  1 )  =  N )
2524oveq2d 6128 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  (
2  x.  ( N  x.  1 ) )  =  ( 2  x.  N ) )
26232timesd 10588 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  (
2  x.  N )  =  ( N  +  N ) )
2725, 26eqtrd 2475 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  (
2  x.  ( N  x.  1 ) )  =  ( N  +  N ) )
2827oveq2d 6128 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  =  ( ( N ^ 2 )  -  ( N  +  N
) ) )
2923sqcld 12027 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  ( N ^ 2 )  e.  CC )
3029, 23, 23subsub4d 9771 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  N
)  -  N )  =  ( ( N ^ 2 )  -  ( N  +  N
) ) )
3128, 30eqtr4d 2478 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  =  ( ( ( N ^ 2 )  -  N )  -  N ) )
32 sq1 11981 . . . . . . . . . . . 12  |-  ( 1 ^ 2 )  =  1
3332a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
1 ^ 2 )  =  1 )
3431, 33oveq12d 6130 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  (
2  x.  ( N  x.  1 ) ) )  +  ( 1 ^ 2 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  N )  +  1 ) )
35 ax-1cn 9361 . . . . . . . . . . 11  |-  1  e.  CC
36 binom2sub 12004 . . . . . . . . . . 11  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
3723, 35, 36sylancl 662 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( N  -  1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  ( 2  x.  ( N  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
3829, 23subcld 9740 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( N ^ 2 )  -  N )  e.  CC )
3935a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  CC )
4038, 23, 39subsubd 9768 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( ( N ^
2 )  -  N
)  -  ( N  -  1 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  N )  +  1 ) )
4134, 37, 403eqtr4d 2485 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( N  -  1 ) ^ 2 )  =  ( ( ( N ^ 2 )  -  N )  -  ( N  -  1
) ) )
4241oveq1d 6127 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( N  - 
1 ) ^ 2 )  +  ( N  -  1 ) )  =  ( ( ( ( N ^ 2 )  -  N )  -  ( N  - 
1 ) )  +  ( N  -  1 ) ) )
43 subcl 9630 . . . . . . . . . 10  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
4423, 35, 43sylancl 662 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  CC )
4538, 44npcand 9744 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( ( N ^ 2 )  -  N )  -  ( N  -  1 ) )  +  ( N  -  1 ) )  =  ( ( N ^ 2 )  -  N ) )
4642, 45eqtrd 2475 . . . . . . 7  |-  ( N  e.  NN  ->  (
( ( N  - 
1 ) ^ 2 )  +  ( N  -  1 ) )  =  ( ( N ^ 2 )  -  N ) )
4746oveq1d 6127 . . . . . 6  |-  ( N  e.  NN  ->  (
( ( ( N  -  1 ) ^
2 )  +  ( N  -  1 ) )  /  2 )  =  ( ( ( N ^ 2 )  -  N )  / 
2 ) )
4822, 47eqtrd 2475 . . . . 5  |-  ( N  e.  NN  ->  sum_ k  e.  ( 1 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
4920, 48eqtrd 2475 . . . 4  |-  ( N  e.  NN  ->  (
0  +  sum_ k  e.  ( ( 0  +  1 ) ... ( N  -  1 ) ) k )  =  ( ( ( N ^ 2 )  -  N )  /  2
) )
509, 49eqtrd 2475 . . 3  |-  ( N  e.  NN  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
51 oveq1 6119 . . . . . . . 8  |-  ( N  =  0  ->  ( N  -  1 )  =  ( 0  -  1 ) )
5251oveq2d 6128 . . . . . . 7  |-  ( N  =  0  ->  (
0 ... ( N  - 
1 ) )  =  ( 0 ... (
0  -  1 ) ) )
53 0re 9407 . . . . . . . . 9  |-  0  e.  RR
54 ltm1 10190 . . . . . . . . 9  |-  ( 0  e.  RR  ->  (
0  -  1 )  <  0 )
5553, 54ax-mp 5 . . . . . . . 8  |-  ( 0  -  1 )  <  0
56 0z 10678 . . . . . . . . 9  |-  0  e.  ZZ
57 peano2zm 10709 . . . . . . . . . 10  |-  ( 0  e.  ZZ  ->  (
0  -  1 )  e.  ZZ )
5856, 57ax-mp 5 . . . . . . . . 9  |-  ( 0  -  1 )  e.  ZZ
59 fzn 11487 . . . . . . . . 9  |-  ( ( 0  e.  ZZ  /\  ( 0  -  1 )  e.  ZZ )  ->  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) ) )
6056, 58, 59mp2an 672 . . . . . . . 8  |-  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) )
6155, 60mpbi 208 . . . . . . 7  |-  ( 0 ... ( 0  -  1 ) )  =  (/)
6252, 61syl6eq 2491 . . . . . 6  |-  ( N  =  0  ->  (
0 ... ( N  - 
1 ) )  =  (/) )
6362sumeq1d 13199 . . . . 5  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  sum_ k  e.  (/)  k )
64 sum0 13219 . . . . 5  |-  sum_ k  e.  (/)  k  =  0
6563, 64syl6eq 2491 . . . 4  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  0 )
66 sq0i 11979 . . . . . . . 8  |-  ( N  =  0  ->  ( N ^ 2 )  =  0 )
67 id 22 . . . . . . . 8  |-  ( N  =  0  ->  N  =  0 )
6866, 67oveq12d 6130 . . . . . . 7  |-  ( N  =  0  ->  (
( N ^ 2 )  -  N )  =  ( 0  -  0 ) )
69 0m0e0 10452 . . . . . . 7  |-  ( 0  -  0 )  =  0
7068, 69syl6eq 2491 . . . . . 6  |-  ( N  =  0  ->  (
( N ^ 2 )  -  N )  =  0 )
7170oveq1d 6127 . . . . 5  |-  ( N  =  0  ->  (
( ( N ^
2 )  -  N
)  /  2 )  =  ( 0  / 
2 ) )
72 2cn 10413 . . . . . 6  |-  2  e.  CC
73 2ne0 10435 . . . . . 6  |-  2  =/=  0
7472, 73div0i 10086 . . . . 5  |-  ( 0  /  2 )  =  0
7571, 74syl6eq 2491 . . . 4  |-  ( N  =  0  ->  (
( ( N ^
2 )  -  N
)  /  2 )  =  0 )
7665, 75eqtr4d 2478 . . 3  |-  ( N  =  0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
7750, 76jaoi 379 . 2  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^ 2 )  -  N )  /  2 ) )
781, 77sylbi 195 1  |-  ( N  e.  NN0  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) k  =  ( ( ( N ^
2 )  -  N
)  /  2 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   (/)c0 3658   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   CCcc 9301   RRcr 9302   0cc0 9303   1c1 9304    + caddc 9306    x. cmul 9308    < clt 9439    - cmin 9616    / cdiv 10014   NNcn 10343   2c2 10392   NN0cn0 10600   ZZcz 10667   ZZ>=cuz 10882   ...cfz 11458   ^cexp 11886   sum_csu 13184
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 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-fz 11459  df-fzo 11570  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-sum 13185
This theorem is referenced by:  birthdaylem3  22369
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