Nuprl Lemma : rabs-of-nonpos
∀[x:ℝ]. |x| = -(x) supposing x ≤ r0
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
req: x = y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
prop: ℙ,
guard: {T}
Lemmas referenced :
rabs-as-rmax,
rmax-req,
rminus_wf,
radd-preserves-rleq,
rleq_functionality,
radd_wf,
rmul_wf,
int-to-real_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
req-iff-rsub-is-0,
itermMinus_wf,
real_term_value_minus_lemma,
req_witness,
rabs_wf,
rleq_wf,
real_wf,
rleq_transitivity,
req_transitivity,
rmul-identity1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
hypothesisEquality,
independent_isectElimination,
because_Cache,
productElimination,
natural_numberEquality,
dependent_functionElimination,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[x:\mBbbR{}]. |x| = -(x) supposing x \mleq{} r0
Date html generated:
2017_10_03-AM-08_30_52
Last ObjectModification:
2017_07_28-AM-07_26_49
Theory : reals
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