Nuprl Lemma : rv-pos-angle-shift
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-pos-angle(n;a;b;c) ⇒ (∀z:ℝ^n. (z ≠ b ⇒ (¬rv-pos-angle(n;a;b;z)) ⇒ rv-pos-angle(n;z;b;c))))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
rv-pos-angle: rv-pos-angle(n;a;b;c),
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real-vec-sep: a ≠ b,
prop: ℙ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
not-rv-pos-angle,
not_wf,
rv-pos-angle_wf,
real-vec-sep_wf,
real-vec_wf,
nat_wf,
rv-pos-angle-implies-separated2,
rv-pos-angle-linearity,
real-vec-add_wf,
real-vec-mul_wf,
real-vec-sub_wf,
rv-pos-angle_functionality,
req-vec_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
productElimination,
because_Cache,
independent_isectElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n.
(rv-pos-angle(n;a;b;c) {}\mRightarrow{} (\mforall{}z:\mBbbR{}\^{}n. (z \mneq{} b {}\mRightarrow{} (\mneg{}rv-pos-angle(n;a;b;z)) {}\mRightarrow{} rv-pos-angle(n;z;b;c))))
Date html generated:
2017_10_03-AM-11_05_05
Last ObjectModification:
2017_03_02-PM-05_12_24
Theory : reals
Home
Index