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Mirrors > Home > MPE Home > Th. List > telgsum | Structured version Visualization version GIF version |
Description: Telescoping finitely supported group sum ranging over nonnegative integers, using implicit substitution. (Contributed by AV, 31-Dec-2019.) |
Ref | Expression |
---|---|
telgsum.b | ⊢ 𝐵 = (Base‘𝐺) |
telgsum.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
telgsum.m | ⊢ − = (-g‘𝐺) |
telgsum.0 | ⊢ 0 = (0g‘𝐺) |
telgsum.f | ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐵) |
telgsum.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
telgsum.u | ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐴 = 0 )) |
telgsum.c | ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐶) |
telgsum.d | ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐷) |
telgsum.e | ⊢ (𝑘 = 0 → 𝐴 = 𝐸) |
Ref | Expression |
---|---|
telgsum | ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷))) = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0) | |
2 | telgsum.c | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐶) | |
3 | 2 | adantl 485 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 = 𝑖) → 𝐴 = 𝐶) |
4 | 1, 3 | csbied 3836 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐶) |
5 | 4 | eqcomd 2742 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐶 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | peano2nn0 12095 | . . . . . . . 8 ⊢ (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℕ0) | |
7 | 6 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑖 + 1) ∈ ℕ0) |
8 | telgsum.d | . . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐷) | |
9 | 8 | adantl 485 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 = (𝑖 + 1)) → 𝐴 = 𝐷) |
10 | 7, 9 | csbied 3836 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐷) |
11 | 10 | eqcomd 2742 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐷 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
12 | 5, 11 | oveq12d 7209 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝐶 − 𝐷) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
13 | 12 | mpteq2dva 5135 | . . 3 ⊢ (𝜑 → (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷)) = (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
14 | 13 | oveq2d 7207 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷))) = (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)))) |
15 | telgsum.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
16 | telgsum.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
17 | telgsum.m | . . 3 ⊢ − = (-g‘𝐺) | |
18 | telgsum.0 | . . 3 ⊢ 0 = (0g‘𝐺) | |
19 | telgsum.f | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐴 ∈ 𝐵) | |
20 | telgsum.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
21 | telgsum.u | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐴 = 0 )) | |
22 | 15, 16, 17, 18, 19, 20, 21 | telgsums 19332 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = ⦋0 / 𝑘⦌𝐴) |
23 | c0ex 10792 | . . . 4 ⊢ 0 ∈ V | |
24 | 23 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
25 | telgsum.e | . . . 4 ⊢ (𝑘 = 0 → 𝐴 = 𝐸) | |
26 | 25 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐴 = 𝐸) |
27 | 24, 26 | csbied 3836 | . 2 ⊢ (𝜑 → ⦋0 / 𝑘⦌𝐴 = 𝐸) |
28 | 14, 22, 27 | 3eqtrd 2775 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷))) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 ⦋csb 3798 class class class wbr 5039 ↦ cmpt 5120 ‘cfv 6358 (class class class)co 7191 0cc0 10694 1c1 10695 + caddc 10697 < clt 10832 ℕ0cn0 12055 Basecbs 16666 0gc0g 16898 Σg cgsu 16899 -gcsg 18321 Abelcabl 19125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-0g 16900 df-gsum 16901 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-cntz 18665 df-cmn 19126 df-abl 19127 |
This theorem is referenced by: (None) |
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