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Mirrors > Home > ILE Home > Th. List > telfsumo2 | GIF version |
Description: Sum of a telescoping series. (Contributed by Mario Carneiro, 2-May-2016.) |
Ref | Expression |
---|---|
telfsumo.1 | ⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵) |
telfsumo.2 | ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶) |
telfsumo.3 | ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) |
telfsumo.4 | ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐸) |
telfsumo.5 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
telfsumo.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
telfsumo2 | ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵) = (𝐸 − 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | telfsumo.1 | . . . 4 ⊢ (𝑘 = 𝑗 → 𝐴 = 𝐵) | |
2 | 1 | negeqd 7957 | . . 3 ⊢ (𝑘 = 𝑗 → -𝐴 = -𝐵) |
3 | telfsumo.2 | . . . 4 ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶) | |
4 | 3 | negeqd 7957 | . . 3 ⊢ (𝑘 = (𝑗 + 1) → -𝐴 = -𝐶) |
5 | telfsumo.3 | . . . 4 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) | |
6 | 5 | negeqd 7957 | . . 3 ⊢ (𝑘 = 𝑀 → -𝐴 = -𝐷) |
7 | telfsumo.4 | . . . 4 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐸) | |
8 | 7 | negeqd 7957 | . . 3 ⊢ (𝑘 = 𝑁 → -𝐴 = -𝐸) |
9 | telfsumo.5 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
10 | telfsumo.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
11 | 10 | negcld 8060 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → -𝐴 ∈ ℂ) |
12 | 2, 4, 6, 8, 9, 11 | telfsumo 11235 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(-𝐵 − -𝐶) = (-𝐷 − -𝐸)) |
13 | 10 | ralrimiva 2505 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
14 | elfzofz 9939 | . . . . 5 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (𝑀...𝑁)) | |
15 | 1 | eleq1d 2208 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
16 | 15 | rspccva 2788 | . . . . 5 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐵 ∈ ℂ) |
17 | 13, 14, 16 | syl2an 287 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐵 ∈ ℂ) |
18 | fzofzp1 10004 | . . . . 5 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (𝑀...𝑁)) | |
19 | 3 | eleq1d 2208 | . . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (𝐴 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
20 | 19 | rspccva 2788 | . . . . 5 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ ∧ (𝑗 + 1) ∈ (𝑀...𝑁)) → 𝐶 ∈ ℂ) |
21 | 13, 18, 20 | syl2an 287 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐶 ∈ ℂ) |
22 | 17, 21 | neg2subd 8090 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (-𝐵 − -𝐶) = (𝐶 − 𝐵)) |
23 | 22 | sumeq2dv 11137 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(-𝐵 − -𝐶) = Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵)) |
24 | 5 | eleq1d 2208 | . . . 4 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐷 ∈ ℂ)) |
25 | eluzfz1 9811 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
26 | 9, 25 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
27 | 24, 13, 26 | rspcdva 2794 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
28 | 7 | eleq1d 2208 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐸 ∈ ℂ)) |
29 | eluzfz2 9812 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | |
30 | 9, 29 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
31 | 28, 13, 30 | rspcdva 2794 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
32 | 27, 31 | neg2subd 8090 | . 2 ⊢ (𝜑 → (-𝐷 − -𝐸) = (𝐸 − 𝐷)) |
33 | 12, 23, 32 | 3eqtr3d 2180 | 1 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵) = (𝐸 − 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 1c1 7621 + caddc 7623 − cmin 7933 -cneg 7934 ℤ≥cuz 9326 ...cfz 9790 ..^cfzo 9919 Σcsu 11122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 |
This theorem is referenced by: telfsum2 11238 |
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