Nuprl Lemma : rminus-rneq-zero
∀x:ℝ. (-(x) ≠ r0 ⇐⇒ x ≠ r0)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rnonzero: rnonzero(x),
exists: ∃x:A. B[x],
member: t ∈ T,
rminus: -(x),
prop: ℙ,
squash: ↓T,
uall: ∀[x:A]. B[x],
real: ℝ,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
true: True
Lemmas referenced :
less_than_wf,
squash_wf,
true_wf,
absval_sym,
absval_wf,
nat_wf,
iff_weakening_equal,
rnonzero_wf,
rminus_wf,
real_wf,
absval-minus,
rnonzero-iff,
rneq_wf,
int-to-real_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
sqequalRule,
hypothesis,
addLevel,
hyp_replacement,
equalitySymmetry,
applyEquality,
lambdaEquality,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
equalityTransitivity,
intEquality,
natural_numberEquality,
equalityEquality,
setElimination,
rename,
minusEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
levelHypothesis,
impliesFunctionality,
dependent_functionElimination
Latex:
\mforall{}x:\mBbbR{}. (-(x) \mneq{} r0 \mLeftarrow{}{}\mRightarrow{} x \mneq{} r0)
Date html generated:
2016_10_26-AM-09_07_59
Last ObjectModification:
2016_07_12-AM-08_17_04
Theory : reals
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