fsumm1 - Intuitionistic Logic Explorer

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Theorem fsumm1 12183

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 9931	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
3	1, 2	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		fzsn 10472	. . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
5	3, 4	syl 14	. . . . 5 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
6	5	ineq2d 3432	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ((𝑀...(𝑁 − 1)) ∩ {𝑁}))
7	3	zred 9768	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8	7	ltm1d 9262	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
9		fzdisj 10457	. . . . 5 ⊢ ((𝑁 − 1) < 𝑁 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ∅)
10	8, 9	syl 14	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ∅)
11	6, 10	eqtr3d 2273	. . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅)
12		eluzel2 9926	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
13	1, 12	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14		peano2zm 9682	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
15	13, 14	syl 14	. . . . . . 7 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
16	13	zcnd 9769	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
17		ax-1cn 8272	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
18		npcan 8535	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀)
19	16, 17, 18	sylancl 417	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
20	19	fveq2d 5699	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘((𝑀 − 1) + 1)) = (ℤ_≥‘𝑀))
21	1, 20	eleqtrrd 2318	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1)))
22		eluzp1m1 9946	. . . . . . 7 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ_≥‘((𝑀 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
23	15, 21, 22	syl2anc 415	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
24		fzsuc2 10486	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ_≥‘(𝑀 − 1))) → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
25	13, 23, 24	syl2anc 415	. . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}))
26	3	zcnd 9769	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
27		npcan 8535	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
28	26, 17, 27	sylancl 417	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
29	28	oveq2d 6101	. . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁))
30	25, 29	eqtr3d 2273	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = (𝑀...𝑁))
31	28	sneqd 3722	. . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁})
32	31	uneq2d 3383	. . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
33	30, 32	eqtr3d 2273	. . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
34	13, 3	fzfigd 10868	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
35		fsumm1.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
36	11, 33, 34, 35	fsumsplit 12174	. 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 2307	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ))
39	35	ralrimiva 2623	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
40		eluzfz2 10436	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
41	1, 40	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
42	38, 39, 41	rspcdva 2934	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ)
43	37	sumsn 12178	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵)
44	1, 42, 43	syl2anc 415	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵)
45	44	oveq2d 6101	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ {𝑁}𝐴) = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))
46	36, 45	eqtrd 2271	1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 ∪ cun 3218 ∩ cin 3219 ∅c0 3520 {csn 3709 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 ℂcc 8177 1c1 8180 + caddc 8182 < clt 8360 − cmin 8497 ℤcz 9644 ℤ_≥cuz 9921 ...cfz 10411 Σcsu 12119

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-n0 9564 df-z 9645 df-uz 9922 df-q 10020 df-rp 10055 df-fz 10412 df-fzo 10550 df-seqfrec 10885 df-exp 10976 df-ihash 11215 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765 df-clim 12045 df-sumdc 12120

This theorem is used by: fzosump1 12184 fsump1 12187 telfsumo 12233 fsumparts 12237 binom1dif 12254 log2ublem2 16084 1sgmprm 16108

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