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Mirrors > Home > MPE Home > Th. List > telfsumo2 | Structured version Visualization version 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 11504 | . . 3 ⊢ (𝑘 = 𝑗 → -𝐴 = -𝐵) |
3 | telfsumo.2 | . . . 4 ⊢ (𝑘 = (𝑗 + 1) → 𝐴 = 𝐶) | |
4 | 3 | negeqd 11504 | . . 3 ⊢ (𝑘 = (𝑗 + 1) → -𝐴 = -𝐶) |
5 | telfsumo.3 | . . . 4 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) | |
6 | 5 | negeqd 11504 | . . 3 ⊢ (𝑘 = 𝑀 → -𝐴 = -𝐷) |
7 | telfsumo.4 | . . . 4 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐸) | |
8 | 7 | negeqd 11504 | . . 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 11608 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → -𝐴 ∈ ℂ) |
12 | 2, 4, 6, 8, 9, 11 | telfsumo 15806 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(-𝐵 − -𝐶) = (-𝐷 − -𝐸)) |
13 | 10 | ralrimiva 3136 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
14 | elfzofz 13702 | . . . . 5 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (𝑀...𝑁)) | |
15 | 1 | eleq1d 2811 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
16 | 15 | rspccva 3607 | . . . . 5 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ ∧ 𝑗 ∈ (𝑀...𝑁)) → 𝐵 ∈ ℂ) |
17 | 13, 14, 16 | syl2an 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐵 ∈ ℂ) |
18 | fzofzp1 13784 | . . . . 5 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝑗 + 1) ∈ (𝑀...𝑁)) | |
19 | 3 | eleq1d 2811 | . . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (𝐴 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
20 | 19 | rspccva 3607 | . . . . 5 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ ∧ (𝑗 + 1) ∈ (𝑀...𝑁)) → 𝐶 ∈ ℂ) |
21 | 13, 18, 20 | syl2an 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐶 ∈ ℂ) |
22 | 17, 21 | neg2subd 11638 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (-𝐵 − -𝐶) = (𝐶 − 𝐵)) |
23 | 22 | sumeq2dv 15707 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(-𝐵 − -𝐶) = Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵)) |
24 | 5 | eleq1d 2811 | . . . 4 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ 𝐷 ∈ ℂ)) |
25 | eluzfz1 13562 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | |
26 | 9, 25 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
27 | 24, 13, 26 | rspcdva 3609 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
28 | 7 | eleq1d 2811 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐸 ∈ ℂ)) |
29 | eluzfz2 13563 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | |
30 | 9, 29 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
31 | 28, 13, 30 | rspcdva 3609 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
32 | 27, 31 | neg2subd 11638 | . 2 ⊢ (𝜑 → (-𝐷 − -𝐸) = (𝐸 − 𝐷)) |
33 | 12, 23, 32 | 3eqtr3d 2774 | 1 ⊢ (𝜑 → Σ𝑗 ∈ (𝑀..^𝑁)(𝐶 − 𝐵) = (𝐸 − 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 1c1 11159 + caddc 11161 − cmin 11494 -cneg 11495 ℤ≥cuz 12874 ...cfz 13538 ..^cfzo 13681 Σcsu 15690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-fz 13539 df-fzo 13682 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-sum 15691 |
This theorem is referenced by: telfsum2 15809 dvfsumle 26045 dvfsumleOLD 26046 dvfsumabs 26048 advlogexp 26682 |
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