Nuprl Lemma : exp-of-mul
∀[x,y:ℤ]. ∀[n:ℕ]. ((x * y)^n = (x^n * y^n) ∈ ℤ)
Proof
Definitions occuring in Statement :
exp: i^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uiff: uiff(P;Q),
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
less_than': less_than'(a;b),
le: A ≤ B,
primtailrec: primtailrec(n;i;b;f),
primrec: primrec(n;b;c),
exp: i^n,
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
exp_step,
false_wf,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
itermMultiply_wf,
intformeq_wf,
multiply-is-int-iff,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
istype-nat,
subtract-1-ge-0,
istype-le,
exp_wf2,
istype-less_than,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties
Rules used in proof :
productElimination,
baseClosed,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
equalitySymmetry,
equalityTransitivity,
intEquality,
cumulativity,
instantiate,
unionElimination,
because_Cache,
Error :isectIsTypeImplies,
Error :dependent_set_memberEquality_alt,
multiplyEquality,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
axiomEquality,
Error :universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
Error :isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
Error :lambdaFormation_alt,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[x,y:\mBbbZ{}]. \mforall{}[n:\mBbbN{}]. ((x * y)\^{}n = (x\^{}n * y\^{}n))
Date html generated:
2019_06_20-PM-02_26_26
Last ObjectModification:
2019_06_19-PM-00_11_31
Theory : num_thy_1
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