Nuprl Lemma : rabs-rinv
∀y:ℝ. (y ≠ r0 ⇒ (|rinv(y)| = rinv(|y|)))
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rabs: |x|,
rinv: rinv(x),
req: x = y,
int-to-real: r(n),
real: ℝ,
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,
rneq: x ≠ y,
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
rabs-neq-zero,
rneq_wf,
int-to-real_wf,
real_wf,
rless-implies-rless,
rminus_wf,
real_term_polynomial,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rsub_wf,
rless_wf,
rabs-of-nonpos,
rleq_weakening_rless,
rabs_wf,
rinv_wf2,
req_functionality,
req_weakening,
rinv_functionality2,
rmul_reverses_rleq_iff,
rmul_wf,
rleq-int,
false_wf,
rleq_functionality,
itermMultiply_wf,
real_term_value_mul_lemma,
req_transitivity,
rmul-rinv,
rmul_preserves_req,
rabs-of-nonneg,
rmul_preserves_rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
addLevel,
unionElimination,
levelHypothesis,
isectElimination,
natural_numberEquality,
inrFormation,
because_Cache,
independent_isectElimination,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_pairFormation
Latex:
\mforall{}y:\mBbbR{}. (y \mneq{} r0 {}\mRightarrow{} (|rinv(y)| = rinv(|y|)))
Date html generated:
2017_10_03-AM-08_37_35
Last ObjectModification:
2017_07_28-AM-07_30_14
Theory : reals
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