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Mirrors > Home > MPE Home > Th. List > Mathboxes > altgsumbc | Structured version Visualization version GIF version |
Description: The sum of binomial coefficients for a fixed positive 𝑁 with alternating signs is zero. Notice that this is not valid for 𝑁 = 0 (since ((-1↑0) · (0C0)) = (1 · 1) = 1). For a proof using Pascal's rule (bcpascm1 44398) instead of the binomial theorem (binom 15184) , see altgsumbcALT 44400. (Contributed by AV, 13-Sep-2019.) |
Ref | Expression |
---|---|
altgsumbc | ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1cnd 10635 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ) | |
2 | negid 10932 | . . . . 5 ⊢ (1 ∈ ℂ → (1 + -1) = 0) | |
3 | 2 | eqcomd 2827 | . . . 4 ⊢ (1 ∈ ℂ → 0 = (1 + -1)) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ → 0 = (1 + -1)) |
5 | 4 | oveq1d 7170 | . 2 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = ((1 + -1)↑𝑁)) |
6 | 0exp 13463 | . 2 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0) | |
7 | 1 | negcld 10983 | . . . 4 ⊢ (𝑁 ∈ ℕ → -1 ∈ ℂ) |
8 | nnnn0 11903 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
9 | binom 15184 | . . . 4 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((1 + -1)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘)))) | |
10 | 1, 7, 8, 9 | syl3anc 1367 | . . 3 ⊢ (𝑁 ∈ ℕ → ((1 + -1)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘)))) |
11 | nnz 12003 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
12 | elfzelz 12907 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ) | |
13 | zsubcl 12023 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑁 − 𝑘) ∈ ℤ) | |
14 | 11, 12, 13 | syl2an 597 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈ ℤ) |
15 | 1exp 13457 | . . . . . . . . 9 ⊢ ((𝑁 − 𝑘) ∈ ℤ → (1↑(𝑁 − 𝑘)) = 1) | |
16 | 14, 15 | syl 17 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (1↑(𝑁 − 𝑘)) = 1) |
17 | 16 | oveq1d 7170 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((1↑(𝑁 − 𝑘)) · (-1↑𝑘)) = (1 · (-1↑𝑘))) |
18 | neg1cn 11750 | . . . . . . . . . 10 ⊢ -1 ∈ ℂ | |
19 | 18 | a1i 11 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → -1 ∈ ℂ) |
20 | elfznn0 12999 | . . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
21 | expcl 13446 | . . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℂ) | |
22 | 19, 20, 21 | syl2an 597 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (-1↑𝑘) ∈ ℂ) |
23 | 22 | mulid2d 10658 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (1 · (-1↑𝑘)) = (-1↑𝑘)) |
24 | 17, 23 | eqtrd 2856 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((1↑(𝑁 − 𝑘)) · (-1↑𝑘)) = (-1↑𝑘)) |
25 | 24 | oveq2d 7171 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘))) = ((𝑁C𝑘) · (-1↑𝑘))) |
26 | bccl 13681 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (𝑁C𝑘) ∈ ℕ0) | |
27 | 8, 12, 26 | syl2an 597 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℕ0) |
28 | 27 | nn0cnd 11956 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ) |
29 | 28, 22 | mulcomd 10661 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · (-1↑𝑘)) = ((-1↑𝑘) · (𝑁C𝑘))) |
30 | 25, 29 | eqtrd 2856 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘))) = ((-1↑𝑘) · (𝑁C𝑘))) |
31 | 30 | sumeq2dv 15059 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘))) = Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘))) |
32 | 10, 31 | eqtrd 2856 | . 2 ⊢ (𝑁 ∈ ℕ → ((1 + -1)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘))) |
33 | 5, 6, 32 | 3eqtr3rd 2865 | 1 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 (class class class)co 7155 ℂcc 10534 0cc0 10536 1c1 10537 + caddc 10539 · cmul 10541 − cmin 10869 -cneg 10870 ℕcn 11637 ℕ0cn0 11896 ℤcz 11980 ...cfz 12891 ↑cexp 13428 Ccbc 13661 Σcsu 15041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-fac 13633 df-bc 13662 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 |
This theorem is referenced by: (None) |
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