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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 44406) instead of the binomial theorem (binom 15188) , see altgsumbcALT 44408. (Contributed by AV, 13-Sep-2019.) |
Ref | Expression |
---|---|
altgsumbc | ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1cnd 10639 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ) | |
2 | negid 10936 | . . . . 5 ⊢ (1 ∈ ℂ → (1 + -1) = 0) | |
3 | 2 | eqcomd 2830 | . . . 4 ⊢ (1 ∈ ℂ → 0 = (1 + -1)) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ → 0 = (1 + -1)) |
5 | 4 | oveq1d 7174 | . 2 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = ((1 + -1)↑𝑁)) |
6 | 0exp 13467 | . 2 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0) | |
7 | 1 | negcld 10987 | . . . 4 ⊢ (𝑁 ∈ ℕ → -1 ∈ ℂ) |
8 | nnnn0 11907 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
9 | binom 15188 | . . . 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 12007 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
12 | elfzelz 12911 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ) | |
13 | zsubcl 12027 | . . . . . . . . . 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 13461 | . . . . . . . . 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 7174 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((1↑(𝑁 − 𝑘)) · (-1↑𝑘)) = (1 · (-1↑𝑘))) |
18 | neg1cn 11754 | . . . . . . . . . 10 ⊢ -1 ∈ ℂ | |
19 | 18 | a1i 11 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → -1 ∈ ℂ) |
20 | elfznn0 13003 | . . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
21 | expcl 13450 | . . . . . . . . 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 10662 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (1 · (-1↑𝑘)) = (-1↑𝑘)) |
24 | 17, 23 | eqtrd 2859 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((1↑(𝑁 − 𝑘)) · (-1↑𝑘)) = (-1↑𝑘)) |
25 | 24 | oveq2d 7175 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘))) = ((𝑁C𝑘) · (-1↑𝑘))) |
26 | bccl 13685 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (𝑁C𝑘) ∈ ℕ0) | |
27 | 8, 12, 26 | syl2an 597 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℕ0) |
28 | 27 | nn0cnd 11960 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ) |
29 | 28, 22 | mulcomd 10665 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · (-1↑𝑘)) = ((-1↑𝑘) · (𝑁C𝑘))) |
30 | 25, 29 | eqtrd 2859 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘))) = ((-1↑𝑘) · (𝑁C𝑘))) |
31 | 30 | sumeq2dv 15063 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘))) = Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘))) |
32 | 10, 31 | eqtrd 2859 | . 2 ⊢ (𝑁 ∈ ℕ → ((1 + -1)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘))) |
33 | 5, 6, 32 | 3eqtr3rd 2868 | 1 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 (class class class)co 7159 ℂcc 10538 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 ℕcn 11641 ℕ0cn0 11900 ℤcz 11984 ...cfz 12895 ↑cexp 13432 Ccbc 13665 Σcsu 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 |
This theorem is referenced by: (None) |
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