Nuprl Lemma : rnexp-req-iff-even
∀n:ℕ+. ∀x,y:ℝ. ((↑isEven(n)) ⇒ (|x| = |y| ⇐⇒ x^n = y^n))
Proof
Definitions occuring in Statement :
rabs: |x|,
rnexp: x^k1,
req: x = y,
real: ℝ,
isEven: isEven(n),
nat_plus: ℕ+,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
rev_implies: P ⇐ Q,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
less_than: a < b,
squash: ↓T,
true: True
Lemmas referenced :
rabs-of-nonneg,
rabs-rmul,
zero-rleq-rabs,
int-to-real_wf,
rleq_wf,
rnexp2,
rnexp_functionality,
rnexp-mul,
req_inversion,
req_transitivity,
req_functionality,
false_wf,
rabs_wf,
iff_wf,
le_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
rnexp_wf,
req_wf,
square-nonneg,
rmul_wf,
less_than_wf,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermMultiply_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rnexp-req-iff,
nat_plus_wf,
real_wf,
isEven_wf,
assert_wf,
int_subtype_base,
subtype_base_sq,
assert-isEven
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
addLevel,
independent_pairFormation,
impliesFunctionality,
dependent_set_memberEquality,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
introduction,
imageMemberEquality,
baseClosed,
allFunctionality,
promote_hyp
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}x,y:\mBbbR{}. ((\muparrow{}isEven(n)) {}\mRightarrow{} (|x| = |y| \mLeftarrow{}{}\mRightarrow{} x\^{}n = y\^{}n))
Date html generated:
2016_05_18-AM-07_29_32
Last ObjectModification:
2016_01_17-AM-02_00_00
Theory : reals
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