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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 47387) instead of the binomial theorem (binom 15802), see altgsumbcALT 47389. (Contributed by AV, 13-Sep-2019.) |
Ref | Expression |
---|---|
altgsumbc | ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1cnd 11233 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ) | |
2 | negid 11531 | . . . . 5 ⊢ (1 ∈ ℂ → (1 + -1) = 0) | |
3 | 2 | eqcomd 2733 | . . . 4 ⊢ (1 ∈ ℂ → 0 = (1 + -1)) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℕ → 0 = (1 + -1)) |
5 | 4 | oveq1d 7429 | . 2 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = ((1 + -1)↑𝑁)) |
6 | 0exp 14088 | . 2 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0) | |
7 | 1 | negcld 11582 | . . . 4 ⊢ (𝑁 ∈ ℕ → -1 ∈ ℂ) |
8 | nnnn0 12503 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
9 | binom 15802 | . . . 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 1369 | . . 3 ⊢ (𝑁 ∈ ℕ → ((1 + -1)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘)))) |
11 | nnz 12603 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
12 | elfzelz 13527 | . . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ) | |
13 | zsubcl 12628 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑁 − 𝑘) ∈ ℤ) | |
14 | 11, 12, 13 | syl2an 595 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (𝑁 − 𝑘) ∈ ℤ) |
15 | 1exp 14082 | . . . . . . . . 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 7429 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((1↑(𝑁 − 𝑘)) · (-1↑𝑘)) = (1 · (-1↑𝑘))) |
18 | neg1cn 12350 | . . . . . . . . . 10 ⊢ -1 ∈ ℂ | |
19 | 18 | a1i 11 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → -1 ∈ ℂ) |
20 | elfznn0 13620 | . . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
21 | expcl 14070 | . . . . . . . . 9 ⊢ ((-1 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (-1↑𝑘) ∈ ℂ) | |
22 | 19, 20, 21 | syl2an 595 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (-1↑𝑘) ∈ ℂ) |
23 | 22 | mullidd 11256 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (1 · (-1↑𝑘)) = (-1↑𝑘)) |
24 | 17, 23 | eqtrd 2767 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((1↑(𝑁 − 𝑘)) · (-1↑𝑘)) = (-1↑𝑘)) |
25 | 24 | oveq2d 7430 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘))) = ((𝑁C𝑘) · (-1↑𝑘))) |
26 | bccl 14307 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (𝑁C𝑘) ∈ ℕ0) | |
27 | 8, 12, 26 | syl2an 595 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℕ0) |
28 | 27 | nn0cnd 12558 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ) |
29 | 28, 22 | mulcomd 11259 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · (-1↑𝑘)) = ((-1↑𝑘) · (𝑁C𝑘))) |
30 | 25, 29 | eqtrd 2767 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘))) = ((-1↑𝑘) · (𝑁C𝑘))) |
31 | 30 | sumeq2dv 15675 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((1↑(𝑁 − 𝑘)) · (-1↑𝑘))) = Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘))) |
32 | 10, 31 | eqtrd 2767 | . 2 ⊢ (𝑁 ∈ ℕ → ((1 + -1)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘))) |
33 | 5, 6, 32 | 3eqtr3rd 2776 | 1 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 (class class class)co 7414 ℂcc 11130 0cc0 11132 1c1 11133 + caddc 11135 · cmul 11137 − cmin 11468 -cneg 11469 ℕcn 12236 ℕ0cn0 12496 ℤcz 12582 ...cfz 13510 ↑cexp 14052 Ccbc 14287 Σcsu 15658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-fac 14259 df-bc 14288 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-sum 15659 |
This theorem is referenced by: (None) |
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