Theorem arisum 12139

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 9463	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		1zzd 9567	. . . . . 6 |- ( N e. NN -> 1 e. ZZ )
3		nnz 9559	. . . . . 6 |- ( N e. NN -> N e. ZZ )
4		elfzelz 10322	. . . . . . . 8 |- ( k e. ( 1 ... N ) -> k e. ZZ )
5	4	zcnd 9664	. . . . . . 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 12086	. . . . 5 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( j + 1 ) )
9		1m1e0 9271	. . . . . . 7 |- ( 1 - 1 ) = 0
10	9	oveq1i 6038	. . . . . 6 |- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
11	10	sumeq1i 12003	. . . . 5 |- sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( j + 1 ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 )
12	8, 11	eqtrdi 2280	. . . 4 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) )
13		elfznn0 10411	. . . . . . . . 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 11084	. . . . . . . 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 9518	. . . . . . . . 9 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. CC )
18		ax-1cn 8185	. . . . . . . . 9 |- 1 e. CC
19		addcom 8375	. . . . . . . . 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 6043	. . . . . . 7 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) _C j ) = ( ( 1 + j ) _C j ) )
22	16, 21	eqtr3d 2266	. . . . . 6 |- ( ( N e. NN /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) = ( ( 1 + j ) _C j ) )
23	22	sumeq2dv 12008	. . . . 5 |- ( N e. NN -> sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( 1 + j ) _C j ) )
24		1nn0 9477	. . . . . 6 |- 1 e. NN0
25		nnm1nn0 9502	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
26		bcxmas 12130	. . . . . 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 2267	. . . 4 |- ( N e. NN -> sum_ j e. ( 0 ... ( N - 1 ) ) ( j + 1 ) = ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) )
29		1cnd 8255	. . . . . . 7 |- ( N e. NN -> 1 e. CC )
30		nncn 9210	. . . . . . 7 |- ( N e. NN -> N e. CC )
31	29, 29, 30	ppncand 8589	. . . . . . 7 |- ( N e. NN -> ( ( 1 + 1 ) + ( N - 1 ) ) = ( 1 + N ) )
32	29, 30, 31	comraddd 8395	. . . . . 6 |- ( N e. NN -> ( ( 1 + 1 ) + ( N - 1 ) ) = ( N + 1 ) )
33	32	oveq1d 6043	. . . . 5 |- ( N e. NN -> ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) = ( ( N + 1 ) _C ( N - 1 ) ) )
34		nnnn0 9468	. . . . . 6 |- ( N e. NN -> N e. NN0 )
35		bcp1m1 11090	. . . . . 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 10922	. . . . . . . . . 10 |- ( N e. CC -> ( N ^ 2 ) = ( N x. N ) )
38	37	eqcomd 2237	. . . . . . . . 9 |- ( N e. CC -> ( N x. N ) = ( N ^ 2 ) )
39		mullid 8237	. . . . . . . . 9 |- ( N e. CC -> ( 1 x. N ) = N )
40	38, 39	oveq12d 6046	. . . . . . . 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 8266	. . . . . 6 |- ( N e. NN -> ( ( N + 1 ) x. N ) = ( ( N ^ 2 ) + N ) )
43	42	oveq1d 6043	. . . . 5 |- ( N e. NN -> ( ( ( N + 1 ) x. N ) / 2 ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
44	33, 36, 43	3eqtrd 2268	. . . 4 |- ( N e. NN -> ( ( ( 1 + 1 ) + ( N - 1 ) ) _C ( N - 1 ) ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
45	12, 28, 44	3eqtrd 2268	. . 3 |- ( N e. NN -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
46		oveq2 6036	. . . . . . 7 |- ( N = 0 -> ( 1 ... N ) = ( 1 ... 0 ) )
47		fz10 10343	. . . . . . 7 |- ( 1 ... 0 ) = (/)
48	46, 47	eqtrdi 2280	. . . . . 6 |- ( N = 0 -> ( 1 ... N ) = (/) )
49	48	sumeq1d 12006	. . . . 5 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = sum_ k e. (/) k )
50		sum0 12029	. . . . 5 |- sum_ k e. (/) k = 0
51	49, 50	eqtrdi 2280	. . . 4 |- ( N = 0 -> sum_ k e. ( 1 ... N ) k = 0 )
52		sq0i 10956	. . . . . . . 8 |- ( N = 0 -> ( N ^ 2 ) = 0 )
53		id 19	. . . . . . . 8 |- ( N = 0 -> N = 0 )
54	52, 53	oveq12d 6046	. . . . . . 7 |- ( N = 0 -> ( ( N ^ 2 ) + N ) = ( 0 + 0 ) )
55		00id 8379	. . . . . . 7 |- ( 0 + 0 ) = 0
56	54, 55	eqtrdi 2280	. . . . . 6 |- ( N = 0 -> ( ( N ^ 2 ) + N ) = 0 )
57	56	oveq1d 6043	. . . . 5 |- ( N = 0 -> ( ( ( N ^ 2 ) + N ) / 2 ) = ( 0 / 2 ) )
58		2cn 9273	. . . . . 6 |- 2 e. CC
59		2ap0 9295	. . . . . 6 |- 2 # 0
60	58, 59	div0api 8985	. . . . 5 |- ( 0 / 2 ) = 0
61	57, 60	eqtrdi 2280	. . . 4 |- ( N = 0 -> ( ( ( N ^ 2 ) + N ) / 2 ) = 0 )
62	51, 61	eqtr4d 2267	. . 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 2202   (/)c0 3496  (class class class)co 6028   CCcc 8090   0cc0 8092   1c1 8093   + caddc 8095   x. cmul 8097   - cmin 8409   / cdiv 8911   NNcn 9202   2c2 9253   NN0cn0 9461   ...cfz 10305   ^cexp 10863   _C cbc 11072   sum_csu 11993

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-fac 11051  df-bc 11073  df-ihash 11101  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-sumdc 11994

This theorem is referenced by:  arisum2  12140