Step
*
of Lemma
combinations-choose
∀[m,n:ℕ]. C(m;n) = (choose(n;m) * (m)!) ∈ ℤ supposing m ≤ n
BY
{ (InstLemma `combinations-formula` []
THEN RepeatFor 2 (ParallelLast)
THEN Auto
THEN ThinTrivial
THEN (SplitOnHypITE -3 THENA Auto)
THEN Auto') }
1
.....truecase.....
1. ∀[n,m:ℕ].
((C(n;m) * (m - n)!) = (m)! ∈ ℤ supposing n ≤ m ∧ (C(n;m) = if n ≤z m then (m)! ÷ (m - n)! else 0 fi ∈ ℤ))
2. m : ℕ
3. ∀[m@0:ℕ]
((C(m;m@0) * (m@0 - m)!) = (m@0)! ∈ ℤ supposing m ≤ m@0
∧ (C(m;m@0) = if m ≤z m@0 then (m@0)! ÷ (m@0 - m)! else 0 fi ∈ ℤ))
4. n : ℕ
5. C(m;n) = ((n)! ÷ (n - m)!) ∈ ℤ
6. m ≤ n
7. (C(m;n) * (n - m)!) = (n)! ∈ ℤ
8. m ≤ n
⊢ C(m;n) = (choose(n;m) * (m)!) ∈ ℤ
Latex:
Latex:
\mforall{}[m,n:\mBbbN{}]. C(m;n) = (choose(n;m) * (m)!) supposing m \mleq{} n
By
Latex:
(InstLemma `combinations-formula` []
THEN RepeatFor 2 (ParallelLast)
THEN Auto
THEN ThinTrivial
THEN (SplitOnHypITE -3 THENA Auto)
THEN Auto')
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