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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 15151. (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 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → 𝑖 ∈ (0...𝑆)) | |
2 | telgsumfz0.a | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐵) | |
3 | 2 | adantl 485 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑆)) ∧ 𝑘 = 𝑖) → 𝐴 = 𝐵) |
4 | 1, 3 | csbied 3864 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐵) |
5 | 4 | eqcomd 2804 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → 𝐵 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | ovexd 7170 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → (𝑖 + 1) ∈ V) | |
7 | telgsumfz0.c | . . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) | |
8 | 7 | adantl 485 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑆)) ∧ 𝑘 = (𝑖 + 1)) → 𝐴 = 𝐶) |
9 | 6, 8 | csbied 3864 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐶) |
10 | 9 | eqcomd 2804 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → 𝐶 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
11 | 5, 10 | oveq12d 7153 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑆)) → (𝐵 − 𝐶) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
12 | 11 | mpteq2dva 5125 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0...𝑆) ↦ (𝐵 − 𝐶)) = (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
13 | 12 | oveq2d 7151 | . 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 19104 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = (⦋0 / 𝑘⦌𝐴 − ⦋(𝑆 + 1) / 𝑘⦌𝐴)) |
20 | c0ex 10624 | . . . . 5 ⊢ 0 ∈ V | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
22 | telgsumfz0.d | . . . . 5 ⊢ (𝑘 = 0 → 𝐴 = 𝐷) | |
23 | 22 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐴 = 𝐷) |
24 | 21, 23 | csbied 3864 | . . 3 ⊢ (𝜑 → ⦋0 / 𝑘⦌𝐴 = 𝐷) |
25 | ovexd 7170 | . . . 4 ⊢ (𝜑 → (𝑆 + 1) ∈ V) | |
26 | telgsumfz0.e | . . . . 5 ⊢ (𝑘 = (𝑆 + 1) → 𝐴 = 𝐸) | |
27 | 26 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = (𝑆 + 1)) → 𝐴 = 𝐸) |
28 | 25, 27 | csbied 3864 | . . 3 ⊢ (𝜑 → ⦋(𝑆 + 1) / 𝑘⦌𝐴 = 𝐸) |
29 | 24, 28 | oveq12d 7153 | . 2 ⊢ (𝜑 → (⦋0 / 𝑘⦌𝐴 − ⦋(𝑆 + 1) / 𝑘⦌𝐴) = (𝐷 − 𝐸)) |
30 | 13, 19, 29 | 3eqtrd 2837 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (0...𝑆) ↦ (𝐵 − 𝐶))) = (𝐷 − 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⦋csb 3828 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 ℕ0cn0 11885 ...cfz 12885 Basecbs 16475 Σg cgsu 16706 -gcsg 18097 Abelcabl 18899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-abl 18901 |
This theorem is referenced by: (None) |
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