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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