Theorem arisum 11267

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 8979	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		1zzd 9081	. . . . . 6 |- ( N e. NN -> 1 e. ZZ )
3		nnz 9073	. . . . . 6 |- ( N e. NN -> N e. ZZ )
4		elfzelz 9806	. . . . . . . 8 |- ( k e. ( 1 ... N ) -> k e. ZZ )
5	4	zcnd 9174	. . . . . . 7 |- ( k e. ( 1 ... N ) -> k e. CC )
6	5	adantl 275	. . . . . 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 11214	. . . . 5 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( j + 1 ) )
9		1m1e0 8789	. . . . . . 7 |- ( 1 - 1 ) = 0
10	9	oveq1i 5784	. . . . . 6 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
11	10	sumeq1i 11132	. . . . 5 |- sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( j + 1 ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 )
12	8, 11	syl6eq 2188	. . . 4 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) )
13		elfznn0 9894	. . . . . . . . 9 |- ( j e. ( 0 ... ( N - 1 ) ) -> j e. NN0 )
14	13	adantl 275	. . . . . . . 8 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. NN0 )
15		bcnp1n 10505	. . . . . . . 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 9032	. . . . . . . . 9 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. CC )
18		ax-1cn 7713	. . . . . . . . 9 |- 1 e. CC
19		addcom 7899	. . . . . . . . 9 |- ( ( j e. CC /\ 1 e. CC ) -> ( j + 1 ) = ( 1 + j ) )
20	17, 18, 19	sylancl 409	. . . . . . . 8 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) = ( 1 + j ) )
21	20	oveq1d 5789	. . . . . . 7 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) _C j ) = ( ( 1 + j ) _C j ) )
22	16, 21	eqtr3d 2174	. . . . . 6 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) = ( ( 1 + j ) _C j ) )
23	22	sumeq2dv 11137	. . . . 5 |- ( N e. NN -> sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( 1 + j ) _C j ) )
24		1nn0 8993	. . . . . 6 |- 1 e. NN0
25		nnm1nn0 9018	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
26		bcxmas 11258	. . . . . 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 410	. . . . 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 2175	. . . 4 |- ( N e. NN -> sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) = ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) )
29		1cnd 7782	. . . . . . 7 |- ( N e. NN -> 1 e. CC )
30		nncn 8728	. . . . . . 7 |- ( N e. NN -> N e. CC )
31	29, 29, 30	ppncand 8113	. . . . . . 7 |- ( N e. NN -> ( ( 1 + 1 ) + ( N - 1 ) ) = ( 1 + N ) )
32	29, 30, 31	comraddd 7919	. . . . . 6 |- ( N e. NN -> ( ( 1 + 1 ) + ( N - 1 ) ) = ( N + 1 ) )
33	32	oveq1d 5789	. . . . 5 |- ( N e. NN -> ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) = ( ( N + 1 ) _C ( N - 1 ) ) )
34		nnnn0 8984	. . . . . 6 |- ( N e. NN -> N e. NN0 )
35		bcp1m1 10511	. . . . . 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 10351	. . . . . . . . . 10 |- ( N e. CC -> ( N ^ 2 ) = ( N x. N ) )
38	37	eqcomd 2145	. . . . . . . . 9 |- ( N e. CC -> ( N x. N ) = ( N ^ 2 ) )
39		mulid2 7764	. . . . . . . . 9 |- ( N e. CC -> ( 1 x. N ) = N )
40	38, 39	oveq12d 5792	. . . . . . . 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 7793	. . . . . 6 |- ( N e. NN -> ( ( N + 1 ) x. N ) = ( ( N ^ 2 ) + N ) )
43	42	oveq1d 5789	. . . . 5 |- ( N e. NN -> ( ( ( N + 1 ) x. N ) / 2 ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
44	33, 36, 43	3eqtrd 2176	. . . 4 |- ( N e. NN -> ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
45	12, 28, 44	3eqtrd 2176	. . 3 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
46		oveq2 5782	. . . . . . 7 |- ( N = 0 -> ( 1 ... N ) = ( 1 ... 0 ) )
47		fz10 9826	. . . . . . 7 |- ( 1 ... 0 ) = (/)
48	46, 47	syl6eq 2188	. . . . . 6 |- ( N = 0 -> ( 1 ... N ) = (/) )
49	48	sumeq1d 11135	. . . . 5 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = sum_ k e. (/) k )
50		sum0 11157	. . . . 5 |- sum_ k e. (/) k = 0
51	49, 50	syl6eq 2188	. . . 4 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = 0 )
52		sq0i 10384	. . . . . . . 8 |- ( N = 0 -> ( N ^ 2 ) = 0 )
53		id 19	. . . . . . . 8 |- ( N = 0 -> N = 0 )
54	52, 53	oveq12d 5792	. . . . . . 7 |- ( N = 0 -> ( ( N ^ 2 ) + N ) = ( 0 + 0 ) )
55		00id 7903	. . . . . . 7 |- ( 0 + 0 ) = 0
56	54, 55	syl6eq 2188	. . . . . 6 |- ( N = 0 -> ( ( N ^ 2 ) + N ) = 0 )
57	56	oveq1d 5789	. . . . 5 |- ( N = 0 -> ( ( ( N ^ 2 ) + N ) / 2 ) = ( 0 / 2 ) )
58		2cn 8791	. . . . . 6 |- 2 e. CC
59		2ap0 8813	. . . . . 6 |- 2 # 0
60	58, 59	div0api 8506	. . . . 5 |- ( 0 / 2 ) = 0
61	57, 60	syl6eq 2188	. . . 4 |- ( N = 0 -> ( ( ( N ^ 2 ) + N ) / 2 ) = 0 )
62	51, 61	eqtr4d 2175	. . 3 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
63	45, 62	jaoi 705	. 2 |- ( ( N e. NN \/ N = 0 ) -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
64	1, 63	sylbi 120	1 |- ( N e. NN0 -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   (/)c0 3363  (class class class)co 5774   CCcc 7618   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   - cmin 7933   / cdiv 8432   NNcn 8720   2c2 8771   NN0cn0 8977   ...cfz 9790   ^cexp 10292   _C cbc 10493   sum_csu 11122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-fac 10472  df-bc 10494  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123

This theorem is referenced by:  arisum2  11268