fsumm1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fsumm1		Structured version Visualization version GIF version

Theorem fsumm1 15463

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 12592	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		fzsn 13298	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
6	5	ineq2d 4146	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ((𝑀...(𝑁 − 1)) ∩ {𝑁}))
7	3	zred 12426	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8	7	ltm1d 11907	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
9		fzdisj 13283	. . . . 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 2780	. . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅)
12		eluzel2 12587	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
13	1, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14		peano2zm 12363	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
16	13	zcnd 12427	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
17		ax-1cn 10929	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
18		npcan 11230	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀)
19	16, 17, 18	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
20	19	fveq2d 6778	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘((𝑀 − 1) + 1)) = (ℤ_≥‘𝑀))
21	1, 20	eleqtrrd 2842	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1)))
22		eluzp1m1 12608	. . . . . . 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 13314	. . . . . 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 12427	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
27		npcan 11230	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
28	26, 17, 27	sylancl 586	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
29	28	oveq2d 7291	. . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
30	25, 29	eqtr3d 2780	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = (𝑀...𝑁))
31	28	sneqd 4573	. . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁})
32	31	uneq2d 4097	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
33	30, 32	eqtr3d 2780	. . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
34		fzfid 13693	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
35		fsumm1.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
36	11, 33, 34, 35	fsumsplit 15453	. 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 2823	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
39	35	ralrimiva 3103	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
40		eluzfz2 13264	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
41	1, 40	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
42	38, 39, 41	rspcdva 3562	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
43	37	sumsn 15458	. . . 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 7291	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ {𝑁}𝐴) = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))
46	36, 45	eqtrd 2778	1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∪ cun 3885 ∩ cin 3886 ∅c0 4256 {csn 4561 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 1c1 10872 + caddc 10874 < clt 11009 − cmin 11205 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 Σcsu 15397

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398

This theorem is referenced by: fzosump1 15464 fsump1 15468 telfsumo 15514 fsumparts 15518 binom1dif 15545 pwdif 15580 bpolysum 15763 bpolydiflem 15764 pwp1fsum 16100 prmreclem4 16620 ovolicc2lem4 24684 dvfsumlem1 25190 abelthlem6 25595 log2ublem2 26097 harmonicbnd4 26160 ftalem1 26222 ftalem5 26226 chpp1 26304 1sgmprm 26347 chtublem 26359 logdivbnd 26704 pntrlog2bndlem1 26725 knoppndvlem15 34706 mettrifi 35915 sticksstones12a 40113 sticksstones12 40114 fzosumm1 40218 stoweidlem17 43558 nnsum4primeseven 45252 nnsum4primesevenALTV 45253

W3C validator