fsumm1 - Metamath Proof Explorer

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Theorem fsumm1 15106

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 12254	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		fzsn 12950	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
6	5	ineq2d 4189	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ((𝑀...(𝑁 − 1)) ∩ {𝑁}))
7	3	zred 12088	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8	7	ltm1d 11572	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
9		fzdisj 12935	. . . . 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 2858	. . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅)
12		eluzel2 12249	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
13	1, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14		peano2zm 12026	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
16	13	zcnd 12089	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
17		ax-1cn 10595	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
18		npcan 10895	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀)
19	16, 17, 18	sylancl 588	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
20	19	fveq2d 6674	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘((𝑀 − 1) + 1)) = (ℤ_≥‘𝑀))
21	1, 20	eleqtrrd 2916	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1)))
22		eluzp1m1 12269	. . . . . . 7 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
23	15, 21, 22	syl2anc 586	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
24		fzsuc2 12966	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1))) → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
25	13, 23, 24	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
26	3	zcnd 12089	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
27		npcan 10895	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
28	26, 17, 27	sylancl 588	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
29	28	oveq2d 7172	. . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
30	25, 29	eqtr3d 2858	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = (𝑀...𝑁))
31	28	sneqd 4579	. . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁})
32	31	uneq2d 4139	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
33	30, 32	eqtr3d 2858	. . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
34		fzfid 13342	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
35		fsumm1.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
36	11, 33, 34, 35	fsumsplit 15097	. 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 2897	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
39	35	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
40		eluzfz2 12916	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
41	1, 40	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
42	38, 39, 41	rspcdva 3625	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
43	37	sumsn 15101	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵)
44	1, 42, 43	syl2anc 586	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵)
45	44	oveq2d 7172	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ {𝑁}𝐴) = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))
46	36, 45	eqtrd 2856	1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 {csn 4567 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 < clt 10675 − cmin 10870 ℤcz 11982 ℤ_≥cuz 12244 ...cfz 12893 Σcsu 15042

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043

This theorem is referenced by: fzosump1 15107 fsump1 15111 telfsumo 15157 fsumparts 15161 binom1dif 15188 pwdif 15223 bpolysum 15407 bpolydiflem 15408 pwp1fsum 15742 prmreclem4 16255 ovolicc2lem4 24121 dvfsumlem1 24623 abelthlem6 25024 log2ublem2 25525 harmonicbnd4 25588 ftalem1 25650 ftalem5 25654 chpp1 25732 1sgmprm 25775 chtublem 25787 logdivbnd 26132 pntrlog2bndlem1 26153 knoppndvlem15 33865 mettrifi 35047 fzosumm1 39146 stoweidlem17 42322 nnsum4primeseven 43985 nnsum4primesevenALTV 43986

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