Nuprl Lemma : rv-pos-angle_functionality
∀n:ℕ. ∀a1,b1,c1,a2,b2,c2:ℝ^n.
(req-vec(n;a1;a2) ⇒ req-vec(n;b1;b2) ⇒ req-vec(n;c1;c2) ⇒ {rv-pos-angle(n;a1;b1;c1) ⇐⇒ rv-pos-angle(n;a2;b2;c2)})
Proof
Definitions occuring in Statement :
rv-pos-angle: rv-pos-angle(n;a;b;c),
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
nat: ℕ,
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rv-pos-angle: rv-pos-angle(n;a;b;c),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
uimplies: b supposing a
Lemmas referenced :
rv-pos-angle_wf,
req-vec_wf,
real-vec_wf,
nat_wf,
rabs_wf,
dot-product_wf,
real-vec-sub_wf,
rmul_wf,
real-vec-norm_wf,
rless_functionality,
rabs_functionality,
dot-product_functionality,
real-vec-sub_functionality,
rmul_functionality,
real-vec-norm_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a1,b1,c1,a2,b2,c2:\mBbbR{}\^{}n.
(req-vec(n;a1;a2)
{}\mRightarrow{} req-vec(n;b1;b2)
{}\mRightarrow{} req-vec(n;c1;c2)
{}\mRightarrow{} \{rv-pos-angle(n;a1;b1;c1) \mLeftarrow{}{}\mRightarrow{} rv-pos-angle(n;a2;b2;c2)\})
Date html generated:
2017_10_03-AM-10_54_40
Last ObjectModification:
2017_03_01-PM-09_16_54
Theory : reals
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