Nuprl Lemma : rpower-nonzero
∀x:ℝ. (x ≠ r0 ⇒ (∀n:ℕ. x^n ≠ r0))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rnexp: x^k1,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
top: Top,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
rnexp_zero_lemma,
rless-int,
rless_wf,
int-to-real_wf,
rneq_wf,
rnexp_wf,
decidable__le,
subtract_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_wf,
real_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rmul_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
rmul-neq-zero,
rneq_functionality,
rnexp-req,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
inrFormation,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
hypothesisEquality,
baseClosed,
isectElimination,
rename,
setElimination,
dependent_set_memberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}x:\mBbbR{}. (x \mneq{} r0 {}\mRightarrow{} (\mforall{}n:\mBbbN{}. x\^{}n \mneq{} r0))
Date html generated:
2017_10_03-AM-08_33_38
Last ObjectModification:
2017_07_28-AM-07_28_27
Theory : reals
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