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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 8114 | . . 3 ⊢ (𝑘 = 𝑗 → -𝐴 = -𝐵) |
3 | telfsumo.2 | . . . 4 ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶) | |
4 | 3 | negeqd 8114 | . . 3 ⊢ (𝑘 = (𝑗 + 1) → -𝐴 = -𝐶) |
5 | telfsumo.3 | . . . 4 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) | |
6 | 5 | negeqd 8114 | . . 3 ⊢ (𝑘 = 𝑀 → -𝐴 = -𝐷) |
7 | telfsumo.4 | . . . 4 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐸) | |
8 | 7 | negeqd 8114 | . . 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 8217 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → -𝐴 ∈ ℂ) |
12 | 2, 4, 6, 8, 9, 11 | telfsumo 11429 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(-𝐵 − -𝐶) = (-𝐷 − -𝐸)) |
13 | 10 | ralrimiva 2543 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
14 | elfzofz 10118 | . . . . 5 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (𝑀...𝑁)) | |
15 | 1 | eleq1d 2239 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
16 | 15 | rspccva 2833 | . . . . 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 10183 | . . . . 5 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (𝑀...𝑁)) | |
19 | 3 | eleq1d 2239 | . . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (𝐴 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
20 | 19 | rspccva 2833 | . . . . 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 8247 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (-𝐵 − -𝐶) = (𝐶 − 𝐵)) |
23 | 22 | sumeq2dv 11331 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(-𝐵 − -𝐶) = Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵)) |
24 | 5 | eleq1d 2239 | . . . 4 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐷 ∈ ℂ)) |
25 | eluzfz1 9987 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
26 | 9, 25 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
27 | 24, 13, 26 | rspcdva 2839 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
28 | 7 | eleq1d 2239 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐸 ∈ ℂ)) |
29 | eluzfz2 9988 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | |
30 | 9, 29 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
31 | 28, 13, 30 | rspcdva 2839 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
32 | 27, 31 | neg2subd 8247 | . 2 ⊢ (𝜑 → (-𝐷 − -𝐸) = (𝐸 − 𝐷)) |
33 | 12, 23, 32 | 3eqtr3d 2211 | 1 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵) = (𝐸 − 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 ∀wral 2448 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 1c1 7775 + caddc 7777 − cmin 8090 -cneg 8091 ℤ≥cuz 9487 ...cfz 9965 ..^cfzo 10098 Σcsu 11316 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317 |
This theorem is referenced by: telfsum2 11432 |
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