bcval3 - Metamath Proof Explorer

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Theorem bcval3 14454

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 14452	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0))
2	1	3adant3 1150	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0))
3		iffalse 4490	. . 3 ⊢ (¬ 𝐾 ∈ (0...𝑁) → if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0) = 0)
4	3	3ad2ant3 1153	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0) = 0)
5	2, 4	eqtrd 2795	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ifcif 4481 ‘cfv 6539 (class class class)co 7421 0cc0 11204 · cmul 11209 − cmin 11545 / cdiv 11977 ℕ₀cn0 12610 ℤcz 12697 ...cfz 13643 !cfa 14421 Ccbc 14450

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-mulcl 11266 ax-i2m1 11272

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-iota 6494 df-fun 6541 df-fv 6547 df-ov 7424 df-oprab 7425 df-mpo 7426 df-bc 14451

This theorem is used by: bcval4 14455 bccmpl 14457 bcval5 14466 bcpasc 14469 bccl 14470 hashbc 14602 binomlem 16004 bcled 43322 bcle2d 43323 bccbc 45423

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