Nuprl Lemma : r2-left-sep
∀a,b,c:ℝ^2. (r2-left(a;b;c) ⇒ b ≠ c)
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
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,
rv-pos-angle: rv-pos-angle(n;a;b;c),
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
iff: P ⇐⇒ Q,
or: P ∨ Q,
real-vec-dist: d(x;y),
real-vec-sep: a ≠ b,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rge: x ≥ y,
guard: {T}
Lemmas referenced :
r2-left-pos-angle,
zero-rleq-rabs,
dot-product_wf,
false_wf,
le_wf,
real-vec-sub_wf,
rmul-is-positive,
real-vec-norm_wf,
real-vec-sep-symmetry,
real-vec-dist-nonneg,
r2-left_wf,
real-vec_wf,
int-to-real_wf,
rabs_wf,
rmul_wf,
real-vec-dist_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rless_transitivity1,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
because_Cache,
productElimination,
unionElimination,
applyEquality,
independent_isectElimination,
voidElimination
Latex:
\mforall{}a,b,c:\mBbbR{}\^{}2. (r2-left(a;b;c) {}\mRightarrow{} b \mneq{} c)
Date html generated:
2017_10_03-AM-11_54_03
Last ObjectModification:
2017_08_11-PM-10_38_50
Theory : reals
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