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Mirrors > Home > MPE Home > Th. List > telgsumfz | Structured version Visualization version GIF version |
Description: Telescoping group sum ranging over a finite set of sequential integers, using implicit substitution, analogous to telfsum 15745. (Contributed by AV, 23-Nov-2019.) |
Ref | Expression |
---|---|
telgsumfz.b | ⊢ 𝐵 = (Base‘𝐺) |
telgsumfz.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
telgsumfz.m | ⊢ − = (-g‘𝐺) |
telgsumfz.n | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
telgsumfz.f | ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 ∈ 𝐵) |
telgsumfz.l | ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐿) |
telgsumfz.c | ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) |
telgsumfz.d | ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) |
telgsumfz.e | ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸) |
Ref | Expression |
---|---|
telgsumfz | ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶))) = (𝐷 − 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝑖 ∈ (𝑀...𝑁)) | |
2 | telgsumfz.l | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐿) | |
3 | 2 | adantl 483 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) ∧ 𝑘 = 𝑖) → 𝐴 = 𝐿) |
4 | 1, 3 | csbied 3929 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐿) |
5 | 4 | eqcomd 2739 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝐿 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | ovexd 7438 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝑖 + 1) ∈ V) | |
7 | telgsumfz.c | . . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) | |
8 | 7 | adantl 483 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) ∧ 𝑘 = (𝑖 + 1)) → 𝐴 = 𝐶) |
9 | 6, 8 | csbied 3929 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐶) |
10 | 9 | eqcomd 2739 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝐶 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
11 | 5, 10 | oveq12d 7421 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝐿 − 𝐶) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
12 | 11 | mpteq2dva 5246 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶)) = (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
13 | 12 | oveq2d 7419 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)))) |
14 | telgsumfz.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
15 | telgsumfz.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
16 | telgsumfz.m | . . 3 ⊢ − = (-g‘𝐺) | |
17 | telgsumfz.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
18 | telgsumfz.f | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 ∈ 𝐵) | |
19 | 14, 15, 16, 17, 18 | telgsumfzs 19848 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = (⦋𝑀 / 𝑘⦌𝐴 − ⦋(𝑁 + 1) / 𝑘⦌𝐴)) |
20 | 17 | elfvexd 6926 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ V) |
21 | telgsumfz.d | . . . . 5 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) | |
22 | 21 | adantl 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐷) |
23 | 20, 22 | csbied 3929 | . . 3 ⊢ (𝜑 → ⦋𝑀 / 𝑘⦌𝐴 = 𝐷) |
24 | ovexd 7438 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ V) | |
25 | telgsumfz.e | . . . . 5 ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸) | |
26 | 25 | adantl 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = (𝑁 + 1)) → 𝐴 = 𝐸) |
27 | 24, 26 | csbied 3929 | . . 3 ⊢ (𝜑 → ⦋(𝑁 + 1) / 𝑘⦌𝐴 = 𝐸) |
28 | 23, 27 | oveq12d 7421 | . 2 ⊢ (𝜑 → (⦋𝑀 / 𝑘⦌𝐴 − ⦋(𝑁 + 1) / 𝑘⦌𝐴) = (𝐷 − 𝐸)) |
29 | 13, 19, 28 | 3eqtrd 2777 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶))) = (𝐷 − 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ⦋csb 3891 ↦ cmpt 5229 ‘cfv 6539 (class class class)co 7403 1c1 11106 + caddc 11108 ℤ≥cuz 12817 ...cfz 13479 Basecbs 17139 Σg cgsu 17381 -gcsg 18816 Abelcabl 19641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-iin 4998 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8141 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-oi 9500 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-n0 12468 df-z 12554 df-uz 12818 df-fz 13480 df-fzo 13623 df-seq 13962 df-hash 14286 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-0g 17382 df-gsum 17383 df-mre 17525 df-mrc 17526 df-acs 17528 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-submnd 18667 df-grp 18817 df-minusg 18818 df-sbg 18819 df-mulg 18944 df-cntz 19174 df-cmn 19642 df-abl 19643 |
This theorem is referenced by: cayhamlem1 22349 |
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