![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 476 | . . . . . . 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 3526 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐶) |
5 | 4 | eqcomd 2616 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐶 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | peano2nn0 11210 | . . . . . . . 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 3526 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐷) |
11 | 10 | eqcomd 2616 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐷 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
12 | 5, 11 | oveq12d 6567 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝐶 − 𝐷) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
13 | 12 | mpteq2dva 4672 | . . 3 ⊢ (𝜑 → (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷)) = (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
14 | 13 | oveq2d 6565 | . 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 18213 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = ⦋0 / 𝑘⦌𝐴) |
23 | c0ex 9913 | . . . 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 3526 | . 2 ⊢ (𝜑 → ⦋0 / 𝑘⦌𝐴 = 𝐸) |
28 | 14, 22, 27 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷))) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⦋csb 3499 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ℕ0cn0 11169 Basecbs 15695 0gc0g 15923 Σg cgsu 15924 -gcsg 17247 Abelcabl 18017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-abl 18019 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |