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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 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0) | |
2 | telgsum.c | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐶) | |
3 | 2 | adantl 484 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 = 𝑖) → 𝐴 = 𝐶) |
4 | 1, 3 | csbied 3918 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐶) |
5 | 4 | eqcomd 2827 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐶 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | peano2nn0 11936 | . . . . . . . 8 ⊢ (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℕ0) | |
7 | 6 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑖 + 1) ∈ ℕ0) |
8 | telgsum.d | . . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐷) | |
9 | 8 | adantl 484 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 = (𝑖 + 1)) → 𝐴 = 𝐷) |
10 | 7, 9 | csbied 3918 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐷) |
11 | 10 | eqcomd 2827 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐷 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
12 | 5, 11 | oveq12d 7173 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝐶 − 𝐷) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
13 | 12 | mpteq2dva 5160 | . . 3 ⊢ (𝜑 → (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷)) = (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
14 | 13 | oveq2d 7171 | . 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 19112 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = ⦋0 / 𝑘⦌𝐴) |
23 | c0ex 10634 | . . . 4 ⊢ 0 ∈ V | |
24 | 23 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
25 | telgsum.e | . . . 4 ⊢ (𝑘 = 0 → 𝐴 = 𝐸) | |
26 | 25 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐴 = 𝐸) |
27 | 24, 26 | csbied 3918 | . 2 ⊢ (𝜑 → ⦋0 / 𝑘⦌𝐴 = 𝐸) |
28 | 14, 22, 27 | 3eqtrd 2860 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷))) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⦋csb 3882 class class class wbr 5065 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 0cc0 10536 1c1 10537 + caddc 10539 < clt 10674 ℕ0cn0 11896 Basecbs 16482 0gc0g 16712 Σg cgsu 16713 -gcsg 18104 Abelcabl 18906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-0g 16714 df-gsum 16715 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-abl 18908 |
This theorem is referenced by: (None) |
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