Nuprl Lemma : r2-perp_wf
∀[x:ℝ^2]. ((r0 < ||x||) ⇒ (r2-perp(x) ∈ {y:ℝ^2| (x⋅y = r0) ∧ (||y|| = r1)} ))
Proof
Definitions occuring in Statement :
r2-perp: r2-perp(x),
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec: ℝ^n,
rless: x < y,
req: x = y,
int-to-real: r(n),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
r2-perp: r2-perp(x),
real-vec: ℝ^n,
int_seg: {i..j-},
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
nat: ℕ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B,
dot-product: x⋅y,
subtract: n - m,
so_lambda: λ2x.t[x],
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_apply: x[s],
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
real: ℝ,
sq_stable: SqStable(P),
eq_int: (i =z j),
rev_uimplies: rev_uimplies(P;Q),
real-vec-norm: ||x||,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
rdiv_wf,
rminus_wf,
lelt_wf,
real-vec-norm_wf,
rless_wf,
false_wf,
le_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,
req_wf,
dot-product_wf,
int-to-real_wf,
real-vec_wf,
rsum_wf,
rmul_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,
radd_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
sq_stable__less_than,
real_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
int_seg_properties,
int_subtype_base,
equal-wf-base,
rmul_preserves_req,
req_weakening,
req_functionality,
rsum-split-first,
radd_functionality,
rsum-single,
uiff_transitivity,
rmul-distrib,
rmul-zero-both,
req_inversion,
rmul-assoc,
radd_comm,
rmul_functionality,
rmul-rdiv-cancel2,
rmul_over_rminus,
rmul_comm,
radd-rminus-both,
rmul-one-both,
rminus_functionality,
req_transitivity,
rminus-rminus,
rsqrt_wf,
dot-product-nonneg,
rleq_wf,
dot-product-comm,
rsqrt_squared,
rleq-int,
rsqrt1,
rsqrt_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
applyEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
inrFormation,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
productEquality,
axiomEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
addEquality,
imageElimination,
setEquality
Latex:
\mforall{}[x:\mBbbR{}\^{}2]. ((r0 < ||x||) {}\mRightarrow{} (r2-perp(x) \mmember{} \{y:\mBbbR{}\^{}2| (x\mcdot{}y = r0) \mwedge{} (||y|| = r1)\} ))
Date html generated:
2017_10_03-PM-00_45_20
Last ObjectModification:
2017_07_28-AM-08_47_13
Theory : reals!model!euclidean!geometry
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