Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > telgsumfz | Structured version Visualization version GIF version |
Description: Telescoping group sum ranging over a finite set of sequential integers, using implicit substitution, analogous to telfsum 15147. (Contributed by AV, 23-Nov-2019.) |
Ref | Expression |
---|---|
telgsumfz.b | ⊢ 𝐵 = (Base‘𝐺) |
telgsumfz.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
telgsumfz.m | ⊢ − = (-g‘𝐺) |
telgsumfz.n | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
telgsumfz.f | ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 ∈ 𝐵) |
telgsumfz.l | ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐿) |
telgsumfz.c | ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) |
telgsumfz.d | ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) |
telgsumfz.e | ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸) |
Ref | Expression |
---|---|
telgsumfz | ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶))) = (𝐷 − 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝑖 ∈ (𝑀...𝑁)) | |
2 | telgsumfz.l | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐿) | |
3 | 2 | adantl 482 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) ∧ 𝑘 = 𝑖) → 𝐴 = 𝐿) |
4 | 1, 3 | csbied 3916 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐿) |
5 | 4 | eqcomd 2824 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝐿 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | ovexd 7180 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝑖 + 1) ∈ V) | |
7 | telgsumfz.c | . . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) | |
8 | 7 | adantl 482 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) ∧ 𝑘 = (𝑖 + 1)) → 𝐴 = 𝐶) |
9 | 6, 8 | csbied 3916 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐶) |
10 | 9 | eqcomd 2824 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝐶 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
11 | 5, 10 | oveq12d 7163 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝐿 − 𝐶) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
12 | 11 | mpteq2dva 5152 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶)) = (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
13 | 12 | oveq2d 7161 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)))) |
14 | telgsumfz.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
15 | telgsumfz.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
16 | telgsumfz.m | . . 3 ⊢ − = (-g‘𝐺) | |
17 | telgsumfz.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
18 | telgsumfz.f | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 ∈ 𝐵) | |
19 | 14, 15, 16, 17, 18 | telgsumfzs 19038 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = (⦋𝑀 / 𝑘⦌𝐴 − ⦋(𝑁 + 1) / 𝑘⦌𝐴)) |
20 | 17 | elfvexd 6697 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ V) |
21 | telgsumfz.d | . . . . 5 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) | |
22 | 21 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐷) |
23 | 20, 22 | csbied 3916 | . . 3 ⊢ (𝜑 → ⦋𝑀 / 𝑘⦌𝐴 = 𝐷) |
24 | ovexd 7180 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ V) | |
25 | telgsumfz.e | . . . . 5 ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸) | |
26 | 25 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = (𝑁 + 1)) → 𝐴 = 𝐸) |
27 | 24, 26 | csbied 3916 | . . 3 ⊢ (𝜑 → ⦋(𝑁 + 1) / 𝑘⦌𝐴 = 𝐸) |
28 | 23, 27 | oveq12d 7163 | . 2 ⊢ (𝜑 → (⦋𝑀 / 𝑘⦌𝐴 − ⦋(𝑁 + 1) / 𝑘⦌𝐴) = (𝐷 − 𝐸)) |
29 | 13, 19, 28 | 3eqtrd 2857 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶))) = (𝐷 − 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ⦋csb 3880 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 1c1 10526 + caddc 10528 ℤ≥cuz 12231 ...cfz 12880 Basecbs 16471 Σg cgsu 16702 -gcsg 18043 Abelcabl 18836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-gsum 16704 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-abl 18838 |
This theorem is referenced by: cayhamlem1 21402 |
Copyright terms: Public domain | W3C validator |