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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 3926 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐶) |
5 | 4 | eqcomd 2732 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐶 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | peano2nn0 12513 | . . . . . . . 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 3926 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐷) |
11 | 10 | eqcomd 2732 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐷 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
12 | 5, 11 | oveq12d 7422 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝐶 − 𝐷) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
13 | 12 | mpteq2dva 5241 | . . 3 ⊢ (𝜑 → (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷)) = (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
14 | 13 | oveq2d 7420 | . 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 19911 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = ⦋0 / 𝑘⦌𝐴) |
23 | c0ex 11209 | . . . 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 3926 | . 2 ⊢ (𝜑 → ⦋0 / 𝑘⦌𝐴 = 𝐸) |
28 | 14, 22, 27 | 3eqtrd 2770 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ ℕ0 ↦ (𝐶 − 𝐷))) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ⦋csb 3888 class class class wbr 5141 ↦ cmpt 5224 ‘cfv 6536 (class class class)co 7404 0cc0 11109 1c1 11110 + caddc 11112 < clt 11249 ℕ0cn0 12473 Basecbs 17151 0gc0g 17392 Σg cgsu 17393 -gcsg 18863 Abelcabl 19699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14294 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-gsum 17395 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-abl 19701 |
This theorem is referenced by: (None) |
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