Nuprl Lemma : exp_mul
∀[i:ℤ]. ∀[n,m:ℕ]. (i^(m * n) = i^m^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 :
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,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
true: True,
nat_plus: ℕ+,
subtract: n - m,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
exp0_lemma,
mul-commutes,
zero-mul,
trivial-equal,
subtract-1-ge-0,
istype-nat,
exp_add,
subtract_wf,
mul_bounds_1a,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
istype-le,
equal_wf,
squash_wf,
true_wf,
istype-universe,
exp_wf2,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
exp_step,
decidable__lt,
add-commutes,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
because_Cache,
intEquality,
dependent_set_memberEquality_alt,
multiplyEquality,
unionElimination,
hyp_replacement,
equalitySymmetry,
applyEquality,
imageElimination,
equalityTransitivity,
instantiate,
universeEquality,
addEquality,
imageMemberEquality,
baseClosed,
minusEquality,
productElimination
Latex:
\mforall{}[i:\mBbbZ{}]. \mforall{}[n,m:\mBbbN{}]. (i\^{}(m * n) = i\^{}m\^{}n)
Date html generated:
2020_05_19-PM-10_01_59
Last ObjectModification:
2019_12_31-AM-10_56_50
Theory : num_thy_1
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