fsumm1 - Metamath Proof Explorer

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Theorem fsumm1 15154

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 12292	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		fzsn 12998	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
6	5	ineq2d 4117	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ((𝑀...(𝑁 − 1)) ∩ {𝑁}))
7	3	zred 12126	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8	7	ltm1d 11610	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
9		fzdisj 12983	. . . . 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 2795	. . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅)
12		eluzel2 12287	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
13	1, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14		peano2zm 12064	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
16	13	zcnd 12127	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
17		ax-1cn 10633	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
18		npcan 10933	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀)
19	16, 17, 18	sylancl 589	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
20	19	fveq2d 6662	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘((𝑀 − 1) + 1)) = (ℤ_≥‘𝑀))
21	1, 20	eleqtrrd 2855	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1)))
22		eluzp1m1 12308	. . . . . . 7 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
23	15, 21, 22	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
24		fzsuc2 13014	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1))) → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
25	13, 23, 24	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
26	3	zcnd 12127	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
27		npcan 10933	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
28	26, 17, 27	sylancl 589	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
29	28	oveq2d 7166	. . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
30	25, 29	eqtr3d 2795	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = (𝑀...𝑁))
31	28	sneqd 4534	. . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁})
32	31	uneq2d 4068	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
33	30, 32	eqtr3d 2795	. . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
34		fzfid 13390	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
35		fsumm1.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
36	11, 33, 34, 35	fsumsplit 15145	. 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 2836	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
39	35	ralrimiva 3113	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
40		eluzfz2 12964	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
41	1, 40	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
42	38, 39, 41	rspcdva 3543	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
43	37	sumsn 15149	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵)
44	1, 42, 43	syl2anc 587	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵)
45	44	oveq2d 7166	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ {𝑁}𝐴) = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))
46	36, 45	eqtrd 2793	1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cun 3856 ∩ cin 3857 ∅c0 4225 {csn 4522 class class class wbr 5032 ‘cfv 6335 (class class class)co 7150 ℂcc 10573 1c1 10576 + caddc 10578 < clt 10713 − cmin 10908 ℤcz 12020 ℤ_≥cuz 12282 ...cfz 12939 Σcsu 15090

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-inf2 9137 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 ax-pre-sup 10653

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-se 5484 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-isom 6344 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7580 df-1st 7693 df-2nd 7694 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-er 8299 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-sup 8939 df-oi 9007 df-card 9401 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-div 11336 df-nn 11675 df-2 11737 df-3 11738 df-n0 11935 df-z 12021 df-uz 12283 df-rp 12431 df-fz 12940 df-fzo 13083 df-seq 13419 df-exp 13480 df-hash 13741 df-cj 14506 df-re 14507 df-im 14508 df-sqrt 14642 df-abs 14643 df-clim 14893 df-sum 15091

This theorem is referenced by: fzosump1 15155 fsump1 15159 telfsumo 15205 fsumparts 15209 binom1dif 15236 pwdif 15271 bpolysum 15455 bpolydiflem 15456 pwp1fsum 15792 prmreclem4 16310 ovolicc2lem4 24220 dvfsumlem1 24725 abelthlem6 25130 log2ublem2 25632 harmonicbnd4 25695 ftalem1 25757 ftalem5 25761 chpp1 25839 1sgmprm 25882 chtublem 25894 logdivbnd 26239 pntrlog2bndlem1 26260 knoppndvlem15 34255 mettrifi 35475 fzosumm1 39724 stoweidlem17 43025 nnsum4primeseven 44685 nnsum4primesevenALTV 44686

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