Nuprl Lemma : r2-left-pos-angle
∀a,b,c:ℝ^2. (r2-left(a;b;c) ⇒ rv-pos-angle(2;a;b;c))
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
rv-pos-angle: rv-pos-angle(n;a;b;c),
real-vec: ℝ^n,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
rv-pos-angle: rv-pos-angle(n;a;b;c),
r2-left: r2-left(p;q;r),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
real-vec: ℝ^n,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
eq_int: (i =z j),
nequal: a ≠ b ∈ T ,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rev_implies: P ⇐ Q,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
itermConstant: "const",
req_int_terms: t1 ≡ t2
Lemmas referenced :
rless_wf,
int-to-real_wf,
r2-det_wf,
real-vec_wf,
false_wf,
le_wf,
dot-product_wf,
real-vec-sub_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
rminus_wf,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_wf,
rless_functionality,
req_weakening,
r2-det-is-dot-product,
radd_wf,
rmul_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
rabs_wf,
real-vec-norm_wf,
r2-dot-product,
rabs_functionality,
square-rless-implies,
rleq_weakening_equal,
real-vec-norm-nonneg,
rleq_wf,
square-nonneg,
rleq_functionality_wrt_implies,
rmul_functionality_wrt_rleq2,
rnexp_wf,
rnexp2-nonneg,
req_inversion,
rabs-rnexp,
req_transitivity,
rnexp-rmul,
rmul_functionality,
real-vec-norm-squared,
rabs-of-nonneg,
rnexp2,
rnexp-rless,
less_than_wf,
rnexp0,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rsub_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
hypothesisEquality,
dependent_set_memberEquality,
independent_pairFormation,
because_Cache,
lambdaEquality,
setElimination,
rename,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
applyEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
functionEquality,
addLevel,
impliesFunctionality,
inlFormation,
productEquality,
int_eqEquality
Latex:
\mforall{}a,b,c:\mBbbR{}\^{}2. (r2-left(a;b;c) {}\mRightarrow{} rv-pos-angle(2;a;b;c))
Date html generated:
2017_10_03-AM-11_49_13
Last ObjectModification:
2017_04_11-PM-05_35_06
Theory : reals
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