Theorem arisum 14592

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 11294	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		1zzd 11408	. . . . . 6 |- ( N e. NN -> 1 e. ZZ )
3		nnz 11399	. . . . . 6 |- ( N e. NN -> N e. ZZ )
4		elfzelz 12342	. . . . . . . 8 |- ( k e. ( 1 ... N ) -> k e. ZZ )
5	4	zcnd 11483	. . . . . . 7 |- ( k e. ( 1 ... N ) -> k e. CC )
6	5	adantl 482	. . . . . 6 |- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> k e. CC )
7		id 22	. . . . . 6 |- ( k = ( j + 1 ) -> k = ( j + 1 ) )
8	2, 2, 3, 6, 7	fsumshftm 14513	. . . . 5 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( j + 1 ) )
9		1m1e0 11089	. . . . . . 7 |- ( 1 - 1 ) = 0
10	9	oveq1i 6660	. . . . . 6 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
11	10	sumeq1i 14428	. . . . 5 |- sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( j + 1 ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 )
12	8, 11	syl6eq 2672	. . . 4 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) )
13		elfznn0 12433	. . . . . . . . 9 |- ( j e. ( 0 ... ( N - 1 ) ) -> j e. NN0 )
14	13	adantl 482	. . . . . . . 8 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. NN0 )
15		bcnp1n 13101	. . . . . . . 8 |- ( j e. NN0 -> ( ( j + 1 ) _C j ) = ( j + 1 ) )
16	14, 15	syl 17	. . . . . . 7 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) _C j ) = ( j + 1 ) )
17	14	nn0cnd 11353	. . . . . . . . 9 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. CC )
18		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
19		addcom 10222	. . . . . . . . 9 |- ( ( j e. CC /\ 1 e. CC ) -> ( j + 1 ) = ( 1 + j ) )
20	17, 18, 19	sylancl 694	. . . . . . . 8 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) = ( 1 + j ) )
21	20	oveq1d 6665	. . . . . . 7 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) _C j ) = ( ( 1 + j ) _C j ) )
22	16, 21	eqtr3d 2658	. . . . . 6 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) = ( ( 1 + j ) _C j ) )
23	22	sumeq2dv 14433	. . . . 5 |- ( N e. NN -> sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( 1 + j ) _C j ) )
24		1nn0 11308	. . . . . 6 |- 1 e. NN0
25		nnm1nn0 11334	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
26		bcxmas 14567	. . . . . 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 695	. . . . 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 2659	. . . 4 |- ( N e. NN -> sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) = ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) )
29		1cnd 10056	. . . . . . . 8 |- ( N e. NN -> 1 e. CC )
30		nncn 11028	. . . . . . . 8 |- ( N e. NN -> N e. CC )
31	29, 29, 30	ppncand 10432	. . . . . . 7 |- ( N e. NN -> ( ( 1 + 1 ) + ( N - 1 ) ) = ( 1 + N ) )
32		addcom 10222	. . . . . . . 8 |- ( ( 1 e. CC /\ N e. CC ) -> ( 1 + N ) = ( N + 1 ) )
33	18, 30, 32	sylancr 695	. . . . . . 7 |- ( N e. NN -> ( 1 + N ) = ( N + 1 ) )
34	31, 33	eqtrd 2656	. . . . . 6 |- ( N e. NN -> ( ( 1 + 1 ) + ( N - 1 ) ) = ( N + 1 ) )
35	34	oveq1d 6665	. . . . 5 |- ( N e. NN -> ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) = ( ( N + 1 ) _C ( N - 1 ) ) )
36		nnnn0 11299	. . . . . 6 |- ( N e. NN -> N e. NN0 )
37		bcp1m1 13107	. . . . . 6 |- ( N e. NN0 -> ( ( N + 1 ) _C ( N - 1 ) ) = ( ( ( N + 1 ) x. N ) / 2 ) )
38	36, 37	syl 17	. . . . 5 |- ( N e. NN -> ( ( N + 1 ) _C ( N - 1 ) ) = ( ( ( N + 1 ) x. N ) / 2 ) )
39	30, 29, 30	adddird 10065	. . . . . . 7 |- ( N e. NN -> ( ( N + 1 ) x. N ) = ( ( N x. N ) + ( 1 x. N ) ) )
40		sqval 12922	. . . . . . . . . 10 |- ( N e. CC -> ( N ^ 2 ) = ( N x. N ) )
41	40	eqcomd 2628	. . . . . . . . 9 |- ( N e. CC -> ( N x. N ) = ( N ^ 2 ) )
42		mulid2 10038	. . . . . . . . 9 |- ( N e. CC -> ( 1 x. N ) = N )
43	41, 42	oveq12d 6668	. . . . . . . 8 |- ( N e. CC -> ( ( N x. N ) + ( 1 x. N ) ) = ( ( N ^ 2 ) + N ) )
44	30, 43	syl 17	. . . . . . 7 |- ( N e. NN -> ( ( N x. N ) + ( 1 x. N ) ) = ( ( N ^ 2 ) + N ) )
45	39, 44	eqtrd 2656	. . . . . 6 |- ( N e. NN -> ( ( N + 1 ) x. N ) = ( ( N ^ 2 ) + N ) )
46	45	oveq1d 6665	. . . . 5 |- ( N e. NN -> ( ( ( N + 1 ) x. N ) / 2 ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
47	35, 38, 46	3eqtrd 2660	. . . 4 |- ( N e. NN -> ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
48	12, 28, 47	3eqtrd 2660	. . 3 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
49		oveq2 6658	. . . . . . 7 |- ( N = 0 -> ( 1 ... N ) = ( 1 ... 0 ) )
50		fz10 12362	. . . . . . 7 |- ( 1 ... 0 ) = (/)
51	49, 50	syl6eq 2672	. . . . . 6 |- ( N = 0 -> ( 1 ... N ) = (/) )
52	51	sumeq1d 14431	. . . . 5 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = sum_ k e. (/) k )
53		sum0 14452	. . . . 5 |- sum_ k e. (/) k = 0
54	52, 53	syl6eq 2672	. . . 4 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = 0 )
55		sq0i 12956	. . . . . . . 8 |- ( N = 0 -> ( N ^ 2 ) = 0 )
56		id 22	. . . . . . . 8 |- ( N = 0 -> N = 0 )
57	55, 56	oveq12d 6668	. . . . . . 7 |- ( N = 0 -> ( ( N ^ 2 ) + N ) = ( 0 + 0 ) )
58		00id 10211	. . . . . . 7 |- ( 0 + 0 ) = 0
59	57, 58	syl6eq 2672	. . . . . 6 |- ( N = 0 -> ( ( N ^ 2 ) + N ) = 0 )
60	59	oveq1d 6665	. . . . 5 |- ( N = 0 -> ( ( ( N ^ 2 ) + N ) / 2 ) = ( 0 / 2 ) )
61		2cn 11091	. . . . . 6 |- 2 e. CC
62		2ne0 11113	. . . . . 6 |- 2 =/= 0
63	61, 62	div0i 10759	. . . . 5 |- ( 0 / 2 ) = 0
64	60, 63	syl6eq 2672	. . . 4 |- ( N = 0 -> ( ( ( N ^ 2 ) + N ) / 2 ) = 0 )
65	54, 64	eqtr4d 2659	. . 3 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
66	48, 65	jaoi 394	. 2 |- ( ( N e. NN \/ N = 0 ) -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
67	1, 66	sylbi 207	1 |- ( N e. NN0 -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ...cfz 12326   ^cexp 12860   _C cbc 13089   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  arisum2  14593