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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 15017. (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 477 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝑖 ∈ (𝑀...𝑁)) | |
2 | telgsumfz.l | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐿) | |
3 | 2 | adantl 474 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) ∧ 𝑘 = 𝑖) → 𝐴 = 𝐿) |
4 | 1, 3 | csbied 3808 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐿) |
5 | 4 | eqcomd 2777 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝐿 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | ovexd 7008 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝑖 + 1) ∈ V) | |
7 | telgsumfz.c | . . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) | |
8 | 7 | adantl 474 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) ∧ 𝑘 = (𝑖 + 1)) → 𝐴 = 𝐶) |
9 | 6, 8 | csbied 3808 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐶) |
10 | 9 | eqcomd 2777 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝐶 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
11 | 5, 10 | oveq12d 6992 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝐿 − 𝐶) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
12 | 11 | mpteq2dva 5018 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶)) = (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
13 | 12 | oveq2d 6990 | . 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 18871 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = (⦋𝑀 / 𝑘⦌𝐴 − ⦋(𝑁 + 1) / 𝑘⦌𝐴)) |
20 | 17 | elfvexd 6531 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ V) |
21 | telgsumfz.d | . . . . 5 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) | |
22 | 21 | adantl 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐷) |
23 | 20, 22 | csbied 3808 | . . 3 ⊢ (𝜑 → ⦋𝑀 / 𝑘⦌𝐴 = 𝐷) |
24 | ovexd 7008 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ V) | |
25 | telgsumfz.e | . . . . 5 ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸) | |
26 | 25 | adantl 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = (𝑁 + 1)) → 𝐴 = 𝐸) |
27 | 24, 26 | csbied 3808 | . . 3 ⊢ (𝜑 → ⦋(𝑁 + 1) / 𝑘⦌𝐴 = 𝐸) |
28 | 23, 27 | oveq12d 6992 | . 2 ⊢ (𝜑 → (⦋𝑀 / 𝑘⦌𝐴 − ⦋(𝑁 + 1) / 𝑘⦌𝐴) = (𝐷 − 𝐸)) |
29 | 13, 19, 28 | 3eqtrd 2811 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶))) = (𝐷 − 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 Vcvv 3408 ⦋csb 3779 ↦ cmpt 5004 ‘cfv 6185 (class class class)co 6974 1c1 10334 + caddc 10336 ℤ≥cuz 12056 ...cfz 12706 Basecbs 16337 Σg cgsu 16568 -gcsg 17905 Abelcabl 18679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 df-fzo 12848 df-seq 13183 df-hash 13504 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-0g 16569 df-gsum 16570 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-mulg 18024 df-cntz 18230 df-cmn 18680 df-abl 18681 |
This theorem is referenced by: cayhamlem1 21193 |
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