Nuprl Lemma : not-real-vec-sep-iff-eq
∀[n:ℕ]. ∀[a,b:ℝ^n]. uiff(¬a ≠ b;req-vec(n;a;b))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A
Definitions unfolded in proof :
real-vec-sep: a ≠ b,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
req-vec: req-vec(n;x;y),
all: ∀x:A. B[x],
real-vec: ℝ^n,
implies: P ⇒ Q,
nat: ℕ,
prop: ℙ,
subtype_rel: A ⊆r B,
not: ¬A,
false: False,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
iff: P ⇐⇒ Q
Lemmas referenced :
req_witness,
int_seg_wf,
not_wf,
rless_wf,
int-to-real_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
req-vec_wf,
real-vec_wf,
nat_wf,
not-rless,
real-vec-dist-identity,
rleq_antisymmetry,
real-vec-dist-nonneg,
nat_plus_properties,
nat_properties,
satisfiable-full-omega-tt,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
rless_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
independent_functionElimination,
hypothesis,
natural_numberEquality,
setElimination,
rename,
setEquality,
lambdaFormation,
voidElimination,
because_Cache,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
addLevel,
impliesFunctionality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a,b:\mBbbR{}\^{}n]. uiff(\mneg{}a \mneq{} b;req-vec(n;a;b))
Date html generated:
2016_10_26-AM-10_30_14
Last ObjectModification:
2016_09_25-AM-01_05_13
Theory : reals
Home
Index