Theorem arisum 12184

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.

Step	Hyp	Ref	Expression
1		elnn0 9498	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		1zzd 9604	. . . . . 6 |- ( N e. NN -> 1 e. ZZ )
3		nnz 9596	. . . . . 6 |- ( N e. NN -> N e. ZZ )
4		elfzelz 10359	. . . . . . . 8 |- ( k e. ( 1 ... N ) -> k e. ZZ )
5	4	zcnd 9701	. . . . . . 7 |- ( k e. ( 1 ... N ) -> k e. CC )
6	5	adantl 277	. . . . . 6 |- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> k e. CC )
7		id 19	. . . . . 6 |- ( k = ( j + 1 ) -> k = ( j + 1 ) )
8	2, 2, 3, 6, 7	fsumshftm 12131	. . . . 5 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( j + 1 ) )
9		1m1e0 9306	. . . . . . 7 |- ( 1 - 1 ) = 0
10	9	oveq1i 6060	. . . . . 6 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
11	10	sumeq1i 12048	. . . . 5 |- sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( j + 1 ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 )
12	8, 11	eqtrdi 2281	. . . 4 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) )
13		elfznn0 10448	. . . . . . . . 9 |- ( j e. ( 0 ... ( N - 1 ) ) -> j e. NN0 )
14	13	adantl 277	. . . . . . . 8 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. NN0 )
15		bcnp1n 11121	. . . . . . . 8 |- ( j e. NN0 -> ( ( j + 1 ) _C j ) = ( j + 1 ) )
16	14, 15	syl 14	. . . . . . 7 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) _C j ) = ( j + 1 ) )
17	14	nn0cnd 9555	. . . . . . . . 9 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. CC )
18		ax-1cn 8220	. . . . . . . . 9 |- 1 e. CC
19		addcom 8410	. . . . . . . . 9 |- ( ( j e. CC /\ 1 e. CC ) -> ( j + 1 ) = ( 1 + j ) )
20	17, 18, 19	sylancl 413	. . . . . . . 8 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) = ( 1 + j ) )
21	20	oveq1d 6065	. . . . . . 7 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) _C j ) = ( ( 1 + j ) _C j ) )
22	16, 21	eqtr3d 2267	. . . . . 6 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) = ( ( 1 + j ) _C j ) )
23	22	sumeq2dv 12053	. . . . 5 |- ( N e. NN -> sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( 1 + j ) _C j ) )
24		1nn0 9512	. . . . . 6 |- 1 e. NN0
25		nnm1nn0 9537	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
26		bcxmas 12175	. . . . . 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 ) )
27	24, 25, 26	sylancr 414	. . . . 5 |- ( N e. NN -> ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( 1 + j ) _C j ) )
28	23, 27	eqtr4d 2268	. . . 4 |- ( N e. NN -> sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) = ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) )
29		1cnd 8290	. . . . . . 7 |- ( N e. NN -> 1 e. CC )
30		nncn 9245	. . . . . . 7 |- ( N e. NN -> N e. CC )
31	29, 29, 30	ppncand 8624	. . . . . . 7 |- ( N e. NN -> ( ( 1 + 1 ) + ( N - 1 ) ) = ( 1 + N ) )
32	29, 30, 31	comraddd 8430	. . . . . 6 |- ( N e. NN -> ( ( 1 + 1 ) + ( N - 1 ) ) = ( N + 1 ) )
33	32	oveq1d 6065	. . . . 5 |- ( N e. NN -> ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) = ( ( N + 1 ) _C ( N - 1 ) ) )
34		nnnn0 9503	. . . . . 6 |- ( N e. NN -> N e. NN0 )
35		bcp1m1 11127	. . . . . 6 |- ( N e. NN0 -> ( ( N + 1 ) _C ( N - 1 ) ) = ( ( ( N + 1 ) x. N ) / 2 ) )
36	34, 35	syl 14	. . . . 5 |- ( N e. NN -> ( ( N + 1 ) _C ( N - 1 ) ) = ( ( ( N + 1 ) x. N ) / 2 ) )
37		sqval 10959	. . . . . . . . . 10 |- ( N e. CC -> ( N ^ 2 ) = ( N x. N ) )
38	37	eqcomd 2238	. . . . . . . . 9 |- ( N e. CC -> ( N x. N ) = ( N ^ 2 ) )
39		mullid 8272	. . . . . . . . 9 |- ( N e. CC -> ( 1 x. N ) = N )
40	38, 39	oveq12d 6068	. . . . . . . 8 |- ( N e. CC -> ( ( N x. N ) + ( 1 x. N ) ) = ( ( N ^ 2 ) + N ) )
41	30, 40	syl 14	. . . . . . 7 |- ( N e. NN -> ( ( N x. N ) + ( 1 x. N ) ) = ( ( N ^ 2 ) + N ) )
42	30, 30, 29, 41	joinlmuladdmuld 8301	. . . . . 6 |- ( N e. NN -> ( ( N + 1 ) x. N ) = ( ( N ^ 2 ) + N ) )
43	42	oveq1d 6065	. . . . 5 |- ( N e. NN -> ( ( ( N + 1 ) x. N ) / 2 ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
44	33, 36, 43	3eqtrd 2269	. . . 4 |- ( N e. NN -> ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
45	12, 28, 44	3eqtrd 2269	. . 3 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
46		oveq2 6058	. . . . . . 7 |- ( N = 0 -> ( 1 ... N ) = ( 1 ... 0 ) )
47		fz10 10380	. . . . . . 7 |- ( 1 ... 0 ) = (/)
48	46, 47	eqtrdi 2281	. . . . . 6 |- ( N = 0 -> ( 1 ... N ) = (/) )
49	48	sumeq1d 12051	. . . . 5 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = sum_ k e. (/) k )
50		sum0 12074	. . . . 5 |- sum_ k e. (/) k = 0
51	49, 50	eqtrdi 2281	. . . 4 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = 0 )
52		sq0i 10993	. . . . . . . 8 |- ( N = 0 -> ( N ^ 2 ) = 0 )
53		id 19	. . . . . . . 8 |- ( N = 0 -> N = 0 )
54	52, 53	oveq12d 6068	. . . . . . 7 |- ( N = 0 -> ( ( N ^ 2 ) + N ) = ( 0 + 0 ) )
55		00id 8414	. . . . . . 7 |- ( 0 + 0 ) = 0
56	54, 55	eqtrdi 2281	. . . . . 6 |- ( N = 0 -> ( ( N ^ 2 ) + N ) = 0 )
57	56	oveq1d 6065	. . . . 5 |- ( N = 0 -> ( ( ( N ^ 2 ) + N ) / 2 ) = ( 0 / 2 ) )
58		2cn 9308	. . . . . 6 |- 2 e. CC
59		2ap0 9330	. . . . . 6 |- 2 # 0
60	58, 59	div0api 9020	. . . . 5 |- ( 0 / 2 ) = 0
61	57, 60	eqtrdi 2281	. . . 4 |- ( N = 0 -> ( ( ( N ^ 2 ) + N ) / 2 ) = 0 )
62	51, 61	eqtr4d 2268	. . 3 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
63	45, 62	jaoi 724	. 2 |- ( ( N e. NN \/ N = 0 ) -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
64	1, 63	sylbi 121	1 |- ( N e. NN0 -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2203   (/)c0 3508  (class class class)co 6050   CCcc 8125   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   - cmin 8444   / cdiv 8946   NNcn 9237   2c2 9288   NN0cn0 9496   ...cfz 10342   ^cexp 10900   _C cbc 11109   sum_csu 12038

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-fac 11088  df-bc 11110  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-sumdc 12039

This theorem is referenced by:  arisum2  12185