Nuprl Lemma : exp-non-zero
∀[n:ℕ]. ∀[x:ℤ]. ¬(x^n = 0 ∈ ℤ) supposing ¬(x = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
exp: i^n,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
ge: i ≥ j ,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
less_than_wf,
equal-wf-base,
exp-is-zero,
equal-wf-T-base,
not_wf,
exp_wf2,
int_subtype_base,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
productEquality,
because_Cache,
baseClosed,
addLevel,
impliesFunctionality,
independent_functionElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbZ{}]. \mneg{}(x\^{}n = 0) supposing \mneg{}(x = 0)
Date html generated:
2017_04_17-AM-09_45_11
Last ObjectModification:
2017_02_27-PM-05_39_57
Theory : num_thy_1
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