bcval3 - Metamath Proof Explorer

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Theorem bcval3 14268

Description: Value of the binomial coefficient, 𝑁 choose 𝐾, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

**Assertion**
Ref	Expression
bcval3	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0)

**Proof of Theorem bcval3**
Step	Hyp	Ref	Expression
1		bcval 14266	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0))
2	1	3adant3 1133	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0))
3		iffalse 4475	. . 3 ⊢ (¬ 𝐾 ∈ (0...𝑁) → if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0) = 0)
4	3	3ad2ant3 1136	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0) = 0)
5	2, 4	eqtrd 2771	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ifcif 4466 ‘cfv 6498 (class class class)co 7367 0cc0 11038 · cmul 11043 − cmin 11377 / cdiv 11807 ℕ₀cn0 12437 ℤcz 12524 ...cfz 13461 !cfa 14235 Ccbc 14264

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-mulcl 11100 ax-i2m1 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6454 df-fun 6500 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-bc 14265

This theorem is referenced by: bcval4 14269 bccmpl 14271 bcval5 14280 bcpasc 14283 bccl 14284 hashbc 14415 binomlem 15794 bcled 42617 bcle2d 42618 bccbc 44772

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