Nuprl Lemma : rneq-rabs
∀a,v:ℝ. (v ≠ a ⇒ -(v) ≠ a ⇒ v ≠ |a|)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rabs: |x|,
rminus: -(x),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rneq: x ≠ y,
or: P ∨ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
guard: {T},
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
less_than: a < b,
less_than': less_than'(a;b),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rneq_wf,
rminus_wf,
real_wf,
rabs_wf,
rless_wf,
rneq_functionality,
req_weakening,
rabs-of-nonneg,
zero-rleq-rabs,
rabs-rless-iff,
rless-implies-rless,
rless_transitivity2,
int-to-real_wf,
rleq_weakening_rless,
rsub_wf,
itermSubtract_wf,
itermVar_wf,
itermMinus_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma,
squash_wf,
true_wf,
rminus-rminus-eq,
iff_weakening_equal,
rless_transitivity1,
rmul_reverses_rless_iff,
rless-int,
rless_functionality,
rmul_wf,
rmul-zero-both,
itermMultiply_wf,
itermConstant_wf,
real_term_value_mul_lemma,
rabs-of-nonpos,
rmul_reverses_rless,
rmul_reverses_rleq_iff,
rleq_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
unionElimination,
because_Cache,
inlFormation,
dependent_functionElimination,
independent_isectElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
natural_numberEquality,
sqequalRule,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
minusEquality,
inrFormation
Latex:
\mforall{}a,v:\mBbbR{}. (v \mneq{} a {}\mRightarrow{} -(v) \mneq{} a {}\mRightarrow{} v \mneq{} |a|)
Date html generated:
2017_10_03-AM-08_39_36
Last ObjectModification:
2017_06_20-AM-11_39_19
Theory : reals
Home
Index