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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 15752. (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 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝑖 ∈ (𝑀...𝑁)) | |
2 | telgsumfz.l | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐿) | |
3 | 2 | adantl 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) ∧ 𝑘 = 𝑖) → 𝐴 = 𝐿) |
4 | 1, 3 | csbied 3924 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐿) |
5 | 4 | eqcomd 2730 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝐿 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | ovexd 7437 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝑖 + 1) ∈ V) | |
7 | telgsumfz.c | . . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) | |
8 | 7 | adantl 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) ∧ 𝑘 = (𝑖 + 1)) → 𝐴 = 𝐶) |
9 | 6, 8 | csbied 3924 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐶) |
10 | 9 | eqcomd 2730 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝐶 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
11 | 5, 10 | oveq12d 7420 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝐿 − 𝐶) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
12 | 11 | mpteq2dva 5239 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶)) = (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
13 | 12 | oveq2d 7418 | . 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 19905 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = (⦋𝑀 / 𝑘⦌𝐴 − ⦋(𝑁 + 1) / 𝑘⦌𝐴)) |
20 | 17 | elfvexd 6921 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ V) |
21 | telgsumfz.d | . . . . 5 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) | |
22 | 21 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐷) |
23 | 20, 22 | csbied 3924 | . . 3 ⊢ (𝜑 → ⦋𝑀 / 𝑘⦌𝐴 = 𝐷) |
24 | ovexd 7437 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ V) | |
25 | telgsumfz.e | . . . . 5 ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸) | |
26 | 25 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = (𝑁 + 1)) → 𝐴 = 𝐸) |
27 | 24, 26 | csbied 3924 | . . 3 ⊢ (𝜑 → ⦋(𝑁 + 1) / 𝑘⦌𝐴 = 𝐸) |
28 | 23, 27 | oveq12d 7420 | . 2 ⊢ (𝜑 → (⦋𝑀 / 𝑘⦌𝐴 − ⦋(𝑁 + 1) / 𝑘⦌𝐴) = (𝐷 − 𝐸)) |
29 | 13, 19, 28 | 3eqtrd 2768 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶))) = (𝐷 − 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ⦋csb 3886 ↦ cmpt 5222 ‘cfv 6534 (class class class)co 7402 1c1 11108 + caddc 11110 ℤ≥cuz 12821 ...cfz 13485 Basecbs 17149 Σg cgsu 17391 -gcsg 18861 Abelcabl 19697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-0g 17392 df-gsum 17393 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 df-cntz 19229 df-cmn 19698 df-abl 19699 |
This theorem is referenced by: cayhamlem1 22712 |
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