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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 15756. (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 3926 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐵) |
5 | 4 | eqcomd 2732 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → 𝐵 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | ovexd 7440 | . . . . . . 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 3926 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐶) |
10 | 9 | eqcomd 2732 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → 𝐶 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
11 | 5, 10 | oveq12d 7423 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → (𝐵 − 𝐶) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
12 | 11 | mpteq2dva 5241 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0...𝑆) ↦ (𝐵 − 𝐶)) = (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
13 | 12 | oveq2d 7421 | . 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 19911 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = (⦋0 / 𝑘⦌𝐴 − ⦋(𝑆 + 1) / 𝑘⦌𝐴)) |
20 | c0ex 11212 | . . . . 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 3926 | . . 3 ⊢ (𝜑 → ⦋0 / 𝑘⦌𝐴 = 𝐷) |
25 | ovexd 7440 | . . . 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 3926 | . . 3 ⊢ (𝜑 → ⦋(𝑆 + 1) / 𝑘⦌𝐴 = 𝐸) |
29 | 24, 28 | oveq12d 7423 | . 2 ⊢ (𝜑 → (⦋0 / 𝑘⦌𝐴 − ⦋(𝑆 + 1) / 𝑘⦌𝐴) = (𝐷 − 𝐸)) |
30 | 13, 19, 29 | 3eqtrd 2770 | 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 3055 Vcvv 3468 ⦋csb 3888 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 0cc0 11112 1c1 11113 + caddc 11115 ℕ0cn0 12476 ...cfz 13490 Basecbs 17153 Σg cgsu 17395 -gcsg 18865 Abelcabl 19701 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-gsum 17397 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-abl 19703 |
This theorem is referenced by: (None) |
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