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Mirrors > Home > MPE Home > Th. List > telgsumfz0 | Structured version Visualization version GIF version |
Description: Telescoping finite group sum ranging over nonnegative integers, using implicit substitution, analogous to telfsum 15747. (Contributed by AV, 23-Nov-2019.) |
Ref | Expression |
---|---|
telgsumfz0.k | ⊢ 𝐾 = (Base‘𝐺) |
telgsumfz0.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
telgsumfz0.m | ⊢ − = (-g‘𝐺) |
telgsumfz0.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
telgsumfz0.f | ⊢ (𝜑 → ∀𝑘 ∈ (0...(𝑆 + 1))𝐴 ∈ 𝐾) |
telgsumfz0.a | ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐵) |
telgsumfz0.c | ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) |
telgsumfz0.d | ⊢ (𝑘 = 0 → 𝐴 = 𝐷) |
telgsumfz0.e | ⊢ (𝑘 = (𝑆 + 1) → 𝐴 = 𝐸) |
Ref | Expression |
---|---|
telgsumfz0 | ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (𝐵 − 𝐶))) = (𝐷 − 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → 𝑖 ∈ (0...𝑆)) | |
2 | telgsumfz0.a | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐵) | |
3 | 2 | adantl 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑆)) ∧ 𝑘 = 𝑖) → 𝐴 = 𝐵) |
4 | 1, 3 | csbied 3923 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐵) |
5 | 4 | eqcomd 2730 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → 𝐵 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | ovexd 7436 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → (𝑖 + 1) ∈ V) | |
7 | telgsumfz0.c | . . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) | |
8 | 7 | adantl 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑆)) ∧ 𝑘 = (𝑖 + 1)) → 𝐴 = 𝐶) |
9 | 6, 8 | csbied 3923 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐶) |
10 | 9 | eqcomd 2730 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → 𝐶 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
11 | 5, 10 | oveq12d 7419 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → (𝐵 − 𝐶) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
12 | 11 | mpteq2dva 5238 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0...𝑆) ↦ (𝐵 − 𝐶)) = (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
13 | 12 | oveq2d 7417 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (𝐵 − 𝐶))) = (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)))) |
14 | telgsumfz0.k | . . 3 ⊢ 𝐾 = (Base‘𝐺) | |
15 | telgsumfz0.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
16 | telgsumfz0.m | . . 3 ⊢ − = (-g‘𝐺) | |
17 | telgsumfz0.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
18 | telgsumfz0.f | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...(𝑆 + 1))𝐴 ∈ 𝐾) | |
19 | 14, 15, 16, 17, 18 | telgsumfz0s 19901 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = (⦋0 / 𝑘⦌𝐴 − ⦋(𝑆 + 1) / 𝑘⦌𝐴)) |
20 | c0ex 11205 | . . . . 5 ⊢ 0 ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
22 | telgsumfz0.d | . . . . 5 ⊢ (𝑘 = 0 → 𝐴 = 𝐷) | |
23 | 22 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐴 = 𝐷) |
24 | 21, 23 | csbied 3923 | . . 3 ⊢ (𝜑 → ⦋0 / 𝑘⦌𝐴 = 𝐷) |
25 | ovexd 7436 | . . . 4 ⊢ (𝜑 → (𝑆 + 1) ∈ V) | |
26 | telgsumfz0.e | . . . . 5 ⊢ (𝑘 = (𝑆 + 1) → 𝐴 = 𝐸) | |
27 | 26 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = (𝑆 + 1)) → 𝐴 = 𝐸) |
28 | 25, 27 | csbied 3923 | . . 3 ⊢ (𝜑 → ⦋(𝑆 + 1) / 𝑘⦌𝐴 = 𝐸) |
29 | 24, 28 | oveq12d 7419 | . 2 ⊢ (𝜑 → (⦋0 / 𝑘⦌𝐴 − ⦋(𝑆 + 1) / 𝑘⦌𝐴) = (𝐷 − 𝐸)) |
30 | 13, 19, 29 | 3eqtrd 2768 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (𝐵 − 𝐶))) = (𝐷 − 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ⦋csb 3885 ↦ cmpt 5221 ‘cfv 6533 (class class class)co 7401 0cc0 11106 1c1 11107 + caddc 11109 ℕ0cn0 12469 ...cfz 13481 Basecbs 17143 Σg cgsu 17385 -gcsg 18855 Abelcabl 19691 |
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 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-0g 17386 df-gsum 17387 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-cntz 19223 df-cmn 19692 df-abl 19693 |
This theorem is referenced by: (None) |
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