bcval3 - Metamath Proof Explorer

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Theorem bcval3 14356

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 14354	. . 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 4498	. . 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 2800	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 1570 ∈ wcel 2146 ifcif 4489 ‘cfv 6540 (class class class)co 7416 0cc0 11111 · cmul 11116 − cmin 11452 / cdiv 11882 ℕ₀cn0 12515 ℤcz 12602 ...cfz 13547 !cfa 14323 Ccbc 14352

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pr 5406 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-mulcl 11173 ax-i2m1 11179

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-iota 6496 df-fun 6542 df-fv 6548 df-ov 7419 df-oprab 7420 df-mpo 7421 df-bc 14353

This theorem is used by: bcval4 14357 bccmpl 14359 bcval5 14368 bcpasc 14371 bccl 14372 hashbc 14504 binomlem 15902 bcled 42978 bcle2d 42979 bccbc 45088

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