Nuprl Lemma : binomial-int
∀[a,b:ℤ]. ∀[n:ℕ]. ((a + b)^n = Σ(choose(n;i) * a^i * b^(n - i) | i < n + 1) ∈ ℤ)
Proof
Definitions occuring in Statement :
exp: i^n,
sum: Σ(f[x] | x < k),
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
choose: choose(n;i)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
integ_dom: IntegDom{i},
int_ring: ℤ-rng,
rng_car: |r|,
pi1: fst(t),
squash: ↓T,
prop: ℙ,
crng: CRng,
rng: Rng,
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
int_iseg: {i...j},
so_apply: x[s],
and: P ∧ Q,
uimplies: b supposing a,
lelt: i ≤ j < k,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
iff: P ⇐⇒ Q,
rng_plus: +r,
pi2: snd(t),
infix_ap: x f y,
rev_implies: P ⇐ Q,
rng_times: *,
cand: A c∧ B
Lemmas referenced :
binomial,
int_ring_wf,
integ_dom_wf,
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
rng_sum-int,
rng_nat_op_wf,
choose_wf,
subtype_rel_sets,
lelt_wf,
le_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
infix_ap_wf,
rng_times_wf,
rng_nexp_wf,
int_seg_subtype_nat,
false_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
int_seg_wf,
iff_weakening_equal,
exp_wf2,
rng_nexp-int,
sum_wf,
nat_wf,
sum_functionality,
rng_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
rng_nat_op-int,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
natural_numberEquality,
addEquality,
intEquality,
productEquality,
independent_isectElimination,
setEquality,
lambdaFormation,
productElimination,
independent_pairFormation,
applyLambdaEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
hyp_replacement,
multiplyEquality,
axiomEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[n:\mBbbN{}]. ((a + b)\^{}n = \mSigma{}(choose(n;i) * a\^{}i * b\^{}(n - i) | i < n + 1))
Date html generated:
2018_05_21-PM-08_27_36
Last ObjectModification:
2017_07_26-PM-05_55_10
Theory : general
Home
Index