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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 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0) | |
2 | telgsum.c | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐶) | |
3 | 2 | adantl 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 = 𝑖) → 𝐴 = 𝐶) |
4 | 1, 3 | csbied 3945 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐶) |
5 | 4 | eqcomd 2740 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐶 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | peano2nn0 12563 | . . . . . . . 8 ⊢ (𝑖 ∈ ℕ0 → (𝑖 + 1) ∈ ℕ0) | |
7 | 6 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑖 + 1) ∈ ℕ0) |
8 | telgsum.d | . . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐷) | |
9 | 8 | adantl 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 = (𝑖 + 1)) → 𝐴 = 𝐷) |
10 | 7, 9 | csbied 3945 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐷) |
11 | 10 | eqcomd 2740 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐷 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
12 | 5, 11 | oveq12d 7448 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝐶 − 𝐷) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
13 | 12 | mpteq2dva 5247 | . . 3 ⊢ (𝜑 → (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷)) = (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
14 | 13 | oveq2d 7446 | . 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 20025 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = ⦋0 / 𝑘⦌𝐴) |
23 | c0ex 11252 | . . . 4 ⊢ 0 ∈ V | |
24 | 23 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
25 | telgsum.e | . . . 4 ⊢ (𝑘 = 0 → 𝐴 = 𝐸) | |
26 | 25 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐴 = 𝐸) |
27 | 24, 26 | csbied 3945 | . 2 ⊢ (𝜑 → ⦋0 / 𝑘⦌𝐴 = 𝐸) |
28 | 14, 22, 27 | 3eqtrd 2778 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷))) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 Vcvv 3477 ⦋csb 3907 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 + caddc 11155 < clt 11292 ℕ0cn0 12523 Basecbs 17244 0gc0g 17485 Σg cgsu 17486 -gcsg 18965 Abelcabl 19813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-0g 17487 df-gsum 17488 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-abl 19815 |
This theorem is referenced by: (None) |
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