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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 15782. (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 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → 𝑖 ∈ (0...𝑆)) | |
2 | telgsumfz0.a | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐵) | |
3 | 2 | adantl 480 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑆)) ∧ 𝑘 = 𝑖) → 𝐴 = 𝐵) |
4 | 1, 3 | csbied 3922 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐵) |
5 | 4 | eqcomd 2731 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → 𝐵 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | ovexd 7451 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → (𝑖 + 1) ∈ V) | |
7 | telgsumfz0.c | . . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) | |
8 | 7 | adantl 480 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑆)) ∧ 𝑘 = (𝑖 + 1)) → 𝐴 = 𝐶) |
9 | 6, 8 | csbied 3922 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐶) |
10 | 9 | eqcomd 2731 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → 𝐶 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
11 | 5, 10 | oveq12d 7434 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → (𝐵 − 𝐶) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
12 | 11 | mpteq2dva 5243 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0...𝑆) ↦ (𝐵 − 𝐶)) = (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
13 | 12 | oveq2d 7432 | . 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 19950 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = (⦋0 / 𝑘⦌𝐴 − ⦋(𝑆 + 1) / 𝑘⦌𝐴)) |
20 | c0ex 11238 | . . . . 5 ⊢ 0 ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
22 | telgsumfz0.d | . . . . 5 ⊢ (𝑘 = 0 → 𝐴 = 𝐷) | |
23 | 22 | adantl 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐴 = 𝐷) |
24 | 21, 23 | csbied 3922 | . . 3 ⊢ (𝜑 → ⦋0 / 𝑘⦌𝐴 = 𝐷) |
25 | ovexd 7451 | . . . 4 ⊢ (𝜑 → (𝑆 + 1) ∈ V) | |
26 | telgsumfz0.e | . . . . 5 ⊢ (𝑘 = (𝑆 + 1) → 𝐴 = 𝐸) | |
27 | 26 | adantl 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = (𝑆 + 1)) → 𝐴 = 𝐸) |
28 | 25, 27 | csbied 3922 | . . 3 ⊢ (𝜑 → ⦋(𝑆 + 1) / 𝑘⦌𝐴 = 𝐸) |
29 | 24, 28 | oveq12d 7434 | . 2 ⊢ (𝜑 → (⦋0 / 𝑘⦌𝐴 − ⦋(𝑆 + 1) / 𝑘⦌𝐴) = (𝐷 − 𝐸)) |
30 | 13, 19, 29 | 3eqtrd 2769 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (𝐵 − 𝐶))) = (𝐷 − 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 Vcvv 3463 ⦋csb 3884 ↦ cmpt 5226 ‘cfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 + caddc 11141 ℕ0cn0 12502 ...cfz 13516 Basecbs 17179 Σg cgsu 17421 -gcsg 18896 Abelcabl 19740 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-0g 17422 df-gsum 17423 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-abl 19742 |
This theorem is referenced by: (None) |
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