Nuprl Lemma : real-binomial
∀[n:ℕ]. ∀[a,b:ℝ]. (a + b^n = Σ{r(choose(n;i)) * a^n - i * b^i | 0≤i≤n})
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rnexp: x^k1,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n,
choose: choose(n;i)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
so_apply: x[s],
int_iseg: {i...j},
guard: {T},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bor: p ∨bq,
btrue: tt,
eq_int: (i =z j),
ycomb: Y,
choose: choose(n;i),
subtract: n - m,
cand: A c∧ B,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
nat_plus: ℕ+,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
it: ⋅,
unit: Unit,
bool: 𝔹,
true: True
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a,b:\mBbbR{}]. (a + b\^{}n = \mSigma{}\{r(choose(n;i)) * a\^{}n - i * b\^{}i | 0\mleq{}i\mleq{}n\})
Date html generated:
2020_05_20-AM-11_13_35
Last ObjectModification:
2020_01_03-AM-00_41_52
Theory : reals
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