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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 15161. (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 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝑖 ∈ (𝑀...𝑁)) | |
2 | telgsumfz.l | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐿) | |
3 | 2 | adantl 484 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) ∧ 𝑘 = 𝑖) → 𝐴 = 𝐿) |
4 | 1, 3 | csbied 3921 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → ⦋𝑖 / 𝑘⦌𝐴 = 𝐿) |
5 | 4 | eqcomd 2829 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝐿 = ⦋𝑖 / 𝑘⦌𝐴) |
6 | ovexd 7193 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝑖 + 1) ∈ V) | |
7 | telgsumfz.c | . . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → 𝐴 = 𝐶) | |
8 | 7 | adantl 484 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) ∧ 𝑘 = (𝑖 + 1)) → 𝐴 = 𝐶) |
9 | 6, 8 | csbied 3921 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → ⦋(𝑖 + 1) / 𝑘⦌𝐴 = 𝐶) |
10 | 9 | eqcomd 2829 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → 𝐶 = ⦋(𝑖 + 1) / 𝑘⦌𝐴) |
11 | 5, 10 | oveq12d 7176 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝐿 − 𝐶) = (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴)) |
12 | 11 | mpteq2dva 5163 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶)) = (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) |
13 | 12 | oveq2d 7174 | . 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 19111 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐴 − ⦋(𝑖 + 1) / 𝑘⦌𝐴))) = (⦋𝑀 / 𝑘⦌𝐴 − ⦋(𝑁 + 1) / 𝑘⦌𝐴)) |
20 | 17 | elfvexd 6706 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ V) |
21 | telgsumfz.d | . . . . 5 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐷) | |
22 | 21 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐷) |
23 | 20, 22 | csbied 3921 | . . 3 ⊢ (𝜑 → ⦋𝑀 / 𝑘⦌𝐴 = 𝐷) |
24 | ovexd 7193 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ V) | |
25 | telgsumfz.e | . . . . 5 ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐸) | |
26 | 25 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = (𝑁 + 1)) → 𝐴 = 𝐸) |
27 | 24, 26 | csbied 3921 | . . 3 ⊢ (𝜑 → ⦋(𝑁 + 1) / 𝑘⦌𝐴 = 𝐸) |
28 | 23, 27 | oveq12d 7176 | . 2 ⊢ (𝜑 → (⦋𝑀 / 𝑘⦌𝐴 − ⦋(𝑁 + 1) / 𝑘⦌𝐴) = (𝐷 − 𝐸)) |
29 | 13, 19, 28 | 3eqtrd 2862 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝐿 − 𝐶))) = (𝐷 − 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ⦋csb 3885 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 1c1 10540 + caddc 10542 ℤ≥cuz 12246 ...cfz 12895 Basecbs 16485 Σg cgsu 16716 -gcsg 18107 Abelcabl 18909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-abl 18911 |
This theorem is referenced by: cayhamlem1 21476 |
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