Nuprl Lemma : rv-pos-angle-implies-separated
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-pos-angle(n;a;b;c) ⇒ a ≠ 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],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
real-vec-sep: a ≠ b,
rv-pos-angle: rv-pos-angle(n;a;b;c),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
real-vec-dist: d(x;y),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
cand: A c∧ B,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
guard: {T}
Lemmas referenced :
rv-pos-angle_wf,
real-vec_wf,
nat_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
int-to-real_wf,
real-vec-sub_wf,
req_weakening,
rless_wf,
rabs_wf,
dot-product_wf,
rmul_wf,
real-vec-norm_wf,
equal_wf,
req_functionality,
real-vec-dist-translation,
rless_functionality,
square-rless-implies,
real-vec-dist-nonneg,
rnexp_wf,
false_wf,
le_wf,
radd_wf,
less_than_wf,
real-vec-norm-diff-squared,
rnexp0,
rnexp-rless,
zero-rleq-rabs,
rnexp2-nonneg,
req_inversion,
rabs-rnexp,
rnexp-rmul,
rabs-of-nonneg,
rsub_wf,
radd-preserves-rleq,
rleq_functionality,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul_preserves_rless,
rless-int,
rnexp2,
rless_transitivity1,
radd-preserves-rless,
radd-non-neg,
rmul_functionality,
rless-implies-rless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
natural_numberEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
productElimination,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
imageMemberEquality,
baseClosed,
productEquality,
computeAll,
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{} a \mneq{} c)
Date html generated:
2017_10_03-AM-11_04_30
Last ObjectModification:
2017_07_28-AM-08_22_30
Theory : reals
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