fsumm1 - Metamath Proof Explorer

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Theorem fsumm1 15674

Description: Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.)

**Hypotheses**
Ref	Expression
fsumm1.1	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
fsumm1.2	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
fsumm1.3	⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵)

**Assertion**
Ref	Expression
fsumm1	⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))

Distinct variable groups: 𝐵,𝑘 𝑘,𝑀 𝑘,𝑁 𝜑,𝑘 Allowed substitution hint: 𝐴(𝑘)

**Proof of Theorem fsumm1**
Step	Hyp	Ref	Expression
1		fsumm1.1	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		eluzelz 12761	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		fzsn 13482	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
6	5	ineq2d 4172	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ((𝑀...(𝑁 − 1)) ∩ {𝑁}))
7	3	zred 12596	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8	7	ltm1d 12074	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
9		fzdisj 13467	. . . . 5 ⊢ ((𝑁 − 1) < 𝑁 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ∅)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ∅)
11	6, 10	eqtr3d 2773	. . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅)
12		eluzel2 12756	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
13	1, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14		peano2zm 12534	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
16	13	zcnd 12597	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
17		ax-1cn 11084	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
18		npcan 11389	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀)
19	16, 17, 18	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
20	19	fveq2d 6838	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘((𝑀 − 1) + 1)) = (ℤ_≥‘𝑀))
21	1, 20	eleqtrrd 2839	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1)))
22		eluzp1m1 12777	. . . . . . 7 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
23	15, 21, 22	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
24		fzsuc2 13498	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1))) → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
25	13, 23, 24	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
26	3	zcnd 12597	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
27		npcan 11389	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
28	26, 17, 27	sylancl 586	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
29	28	oveq2d 7374	. . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
30	25, 29	eqtr3d 2773	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = (𝑀...𝑁))
31	28	sneqd 4592	. . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁})
32	31	uneq2d 4120	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
33	30, 32	eqtr3d 2773	. . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
34		fzfid 13896	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
35		fsumm1.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
36	11, 33, 34, 35	fsumsplit 15664	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ {𝑁}𝐴))
37		fsumm1.3	. . . . . 6 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵)
38	37	eleq1d 2821	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
39	35	ralrimiva 3128	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
40		eluzfz2 13448	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
41	1, 40	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
42	38, 39, 41	rspcdva 3577	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
43	37	sumsn 15669	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵)
44	1, 42, 43	syl2anc 584	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵)
45	44	oveq2d 7374	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ {𝑁}𝐴) = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))
46	36, 45	eqtrd 2771	1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∪ cun 3899 ∩ cin 3900 ∅c0 4285 {csn 4580 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 1c1 11027 + caddc 11029 < clt 11166 − cmin 11364 ℤcz 12488 ℤ_≥cuz 12751 ...cfz 13423 Σcsu 15609

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-z 12489 df-uz 12752 df-rp 12906 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610

This theorem is referenced by: fzosump1 15675 fsump1 15679 telfsumo 15725 fsumparts 15729 binom1dif 15756 pwdif 15791 bpolysum 15976 bpolydiflem 15977 pwp1fsum 16318 prmreclem4 16847 ovolicc2lem4 25477 dvfsumlem1 25988 abelthlem6 26402 log2ublem2 26913 harmonicbnd4 26977 ftalem1 27039 ftalem5 27043 chpp1 27121 1sgmprm 27166 chtublem 27178 logdivbnd 27523 pntrlog2bndlem1 27544 knoppndvlem15 36726 mettrifi 37958 sticksstones12a 42411 sticksstones12 42412 fzosumm1 42505 fz1sump1 42565 stoweidlem17 46261 nnsum4primeseven 48046 nnsum4primesevenALTV 48047

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