Nuprl Lemma : rv-pos-angle-linearity
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-pos-angle(n;a;b;c) ⇒ (∀t:ℝ. ((r0 < |t|) ⇒ rv-pos-angle(n;b + t*a - b;b;c))))
Proof
Definitions occuring in Statement :
rv-pos-angle: rv-pos-angle(n;a;b;c),
real-vec-mul: a*X,
real-vec-sub: X - Y,
real-vec-add: X + Y,
real-vec: ℝ^n,
rless: x < y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rv-pos-angle: rv-pos-angle(n;a;b;c),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
real-vec-sub: X - Y,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
and: P ∧ Q,
rsub: x - y,
uimplies: b supposing a,
real-vec: ℝ^n,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rless_wf,
int-to-real_wf,
rabs_wf,
real_wf,
rv-pos-angle_wf,
real-vec_wf,
nat_wf,
equal_wf,
real-vec-sub_wf,
real-vec-mul_wf,
int_seg_wf,
radd-rminus-assoc,
radd_comm,
radd_functionality,
radd-assoc,
req_inversion,
req_functionality,
uiff_transitivity,
req_weakening,
rminus_wf,
radd_wf,
req_wf,
dot-product_wf,
rmul_wf,
real-vec-norm_wf,
real-vec-add_wf,
rless_functionality,
req_transitivity,
rabs_functionality,
dot-product-linearity2,
rabs-rmul,
rmul_functionality,
real-vec-norm-mul,
dot-product_functionality,
req-vec_weakening,
real-vec-norm_functionality,
rmul_preserves_rless,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rsub_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
rename,
setElimination,
sqequalRule,
productElimination,
independent_isectElimination,
because_Cache,
applyEquality,
addLevel,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n.
(rv-pos-angle(n;a;b;c) {}\mRightarrow{} (\mforall{}t:\mBbbR{}. ((r0 < |t|) {}\mRightarrow{} rv-pos-angle(n;b + t*a - b;b;c))))
Date html generated:
2017_10_03-AM-10_58_19
Last ObjectModification:
2017_07_28-AM-08_21_43
Theory : reals
Home
Index