Theorem arisum 11644

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 9245	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		1zzd 9347	. . . . . 6 |- ( N e. NN -> 1 e. ZZ )
3		nnz 9339	. . . . . 6 |- ( N e. NN -> N e. ZZ )
4		elfzelz 10094	. . . . . . . 8 |- ( k e. ( 1 ... N ) -> k e. ZZ )
5	4	zcnd 9443	. . . . . . 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 11591	. . . . 5 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( j + 1 ) )
9		1m1e0 9053	. . . . . . 7 |- ( 1 - 1 ) = 0
10	9	oveq1i 5929	. . . . . 6 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
11	10	sumeq1i 11509	. . . . 5 |- sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( j + 1 ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 )
12	8, 11	eqtrdi 2242	. . . 4 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) )
13		elfznn0 10183	. . . . . . . . 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 10833	. . . . . . . 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 9298	. . . . . . . . 9 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. CC )
18		ax-1cn 7967	. . . . . . . . 9 |- 1 e. CC
19		addcom 8158	. . . . . . . . 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 5934	. . . . . . 7 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) _C j ) = ( ( 1 + j ) _C j ) )
22	16, 21	eqtr3d 2228	. . . . . 6 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) = ( ( 1 + j ) _C j ) )
23	22	sumeq2dv 11514	. . . . 5 |- ( N e. NN -> sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( 1 + j ) _C j ) )
24		1nn0 9259	. . . . . 6 |- 1 e. NN0
25		nnm1nn0 9284	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
26		bcxmas 11635	. . . . . 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 2229	. . . 4 |- ( N e. NN -> sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) = ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) )
29		1cnd 8037	. . . . . . 7 |- ( N e. NN -> 1 e. CC )
30		nncn 8992	. . . . . . 7 |- ( N e. NN -> N e. CC )
31	29, 29, 30	ppncand 8372	. . . . . . 7 |- ( N e. NN -> ( ( 1 + 1 ) + ( N - 1 ) ) = ( 1 + N ) )
32	29, 30, 31	comraddd 8178	. . . . . 6 |- ( N e. NN -> ( ( 1 + 1 ) + ( N - 1 ) ) = ( N + 1 ) )
33	32	oveq1d 5934	. . . . 5 |- ( N e. NN -> ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) = ( ( N + 1 ) _C ( N - 1 ) ) )
34		nnnn0 9250	. . . . . 6 |- ( N e. NN -> N e. NN0 )
35		bcp1m1 10839	. . . . . 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 10671	. . . . . . . . . 10 |- ( N e. CC -> ( N ^ 2 ) = ( N x. N ) )
38	37	eqcomd 2199	. . . . . . . . 9 |- ( N e. CC -> ( N x. N ) = ( N ^ 2 ) )
39		mullid 8019	. . . . . . . . 9 |- ( N e. CC -> ( 1 x. N ) = N )
40	38, 39	oveq12d 5937	. . . . . . . 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 8049	. . . . . 6 |- ( N e. NN -> ( ( N + 1 ) x. N ) = ( ( N ^ 2 ) + N ) )
43	42	oveq1d 5934	. . . . 5 |- ( N e. NN -> ( ( ( N + 1 ) x. N ) / 2 ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
44	33, 36, 43	3eqtrd 2230	. . . 4 |- ( N e. NN -> ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
45	12, 28, 44	3eqtrd 2230	. . 3 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
46		oveq2 5927	. . . . . . 7 |- ( N = 0 -> ( 1 ... N ) = ( 1 ... 0 ) )
47		fz10 10115	. . . . . . 7 |- ( 1 ... 0 ) = (/)
48	46, 47	eqtrdi 2242	. . . . . 6 |- ( N = 0 -> ( 1 ... N ) = (/) )
49	48	sumeq1d 11512	. . . . 5 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = sum_ k e. (/) k )
50		sum0 11534	. . . . 5 |- sum_ k e. (/) k = 0
51	49, 50	eqtrdi 2242	. . . 4 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = 0 )
52		sq0i 10705	. . . . . . . 8 |- ( N = 0 -> ( N ^ 2 ) = 0 )
53		id 19	. . . . . . . 8 |- ( N = 0 -> N = 0 )
54	52, 53	oveq12d 5937	. . . . . . 7 |- ( N = 0 -> ( ( N ^ 2 ) + N ) = ( 0 + 0 ) )
55		00id 8162	. . . . . . 7 |- ( 0 + 0 ) = 0
56	54, 55	eqtrdi 2242	. . . . . 6 |- ( N = 0 -> ( ( N ^ 2 ) + N ) = 0 )
57	56	oveq1d 5934	. . . . 5 |- ( N = 0 -> ( ( ( N ^ 2 ) + N ) / 2 ) = ( 0 / 2 ) )
58		2cn 9055	. . . . . 6 |- 2 e. CC
59		2ap0 9077	. . . . . 6 |- 2 # 0
60	58, 59	div0api 8767	. . . . 5 |- ( 0 / 2 ) = 0
61	57, 60	eqtrdi 2242	. . . 4 |- ( N = 0 -> ( ( ( N ^ 2 ) + N ) / 2 ) = 0 )
62	51, 61	eqtr4d 2229	. . 3 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
63	45, 62	jaoi 717	. 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 709   = wceq 1364   e. wcel 2164   (/)c0 3447  (class class class)co 5919   CCcc 7872   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   - cmin 8192   / cdiv 8693   NNcn 8984   2c2 9035   NN0cn0 9243   ...cfz 10077   ^cexp 10612   _C cbc 10821   sum_csu 11499

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-fac 10800  df-bc 10822  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500

This theorem is referenced by:  arisum2  11645