Nuprl Lemma : r2-det-nonzero
∀p,q,r:ℝ^2. (rv-pos-angle(2;p;q;r) ⇒ |pqr| ≠ r0)
Proof
Definitions occuring in Statement :
r2-det: |pqr|,
rv-pos-angle: rv-pos-angle(n;a;b;c),
real-vec: ℝ^n,
rneq: x ≠ y,
int-to-real: r(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,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
real-vec-sub: X - Y,
r2-det: |pqr|,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
uimplies: b supposing a,
rsub: x - y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
dot-product: x⋅y,
subtract: n - m,
so_lambda: λ2x.t[x],
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
lt_int: i <z j,
eq_int: (i =z j),
rneq: x ≠ y
Lemmas referenced :
rv-pos-angle_wf,
false_wf,
le_wf,
real-vec_wf,
lelt_wf,
real_wf,
equal_wf,
req_wf,
rsub_wf,
radd_wf,
rmul_wf,
int-to-real_wf,
rminus_wf,
req_weakening,
uiff_transitivity,
req_functionality,
radd_functionality,
rminus_functionality,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
rmul_comm,
rminus-radd,
rmul_functionality,
rminus-rminus,
req_inversion,
radd-assoc,
radd-ac,
radd_comm,
rminus-as-rmul,
rmul-identity1,
rmul-distrib2,
radd-int,
rmul-zero-both,
radd-zero-both,
r2-det_wf,
real-vec-sub_wf,
rless_wf,
rabs_wf,
dot-product_wf,
real-vec-norm_wf,
rneq_functionality,
square-nonzero,
rnexp-rless,
zero-rleq-rabs,
less_than_wf,
rnexp_wf,
rnexp2-nonneg,
rless_functionality,
rabs-rnexp,
rnexp-rmul,
real-vec-norm-squared,
rabs-of-nonneg,
rnexp2,
rsum_wf,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
int_seg_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
eq_int_wf,
assert_of_eq_int,
int_subtype_base,
neg_assert_of_eq_int,
subtract_wf,
subtract-add-cancel,
rsum_unroll,
rsum_single,
radd-preserves-rless,
rmul-assoc,
rmul-ac
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
hypothesis,
hypothesisEquality,
applyEquality,
because_Cache,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
minusEquality,
addEquality,
independent_isectElimination,
productElimination,
lambdaEquality,
setElimination,
rename,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
inrFormation,
addLevel
Latex:
\mforall{}p,q,r:\mBbbR{}\^{}2. (rv-pos-angle(2;p;q;r) {}\mRightarrow{} |pqr| \mneq{} r0)
Date html generated:
2017_10_03-AM-11_47_05
Last ObjectModification:
2017_04_11-PM-05_33_14
Theory : reals
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