Nuprl Lemma : r2-dot-product-eq-0-iff-perp
∀x:ℝ^2. ((r0 < ||x||) 
⇒ (∀y:ℝ^2. (y⋅x = r0 
⇐⇒ ∃t:ℝ. req-vec(2;y;t*r2-perp(x)))))
Proof
Definitions occuring in Statement : 
r2-perp: r2-perp(x)
, 
real-vec-norm: ||x||
, 
dot-product: x⋅y
, 
real-vec-mul: a*X
, 
req-vec: req-vec(n;x;y)
, 
real-vec: ℝ^n
, 
rless: x < y
, 
req: x = y
, 
int-to-real: r(n)
, 
real: ℝ
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
implies: P 
⇒ Q
, 
natural_number: $n
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
nat: ℕ
, 
rless: x < y
, 
sq_exists: ∃x:A [B[x]]
, 
nat_plus: ℕ+
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
uimplies: b supposing a
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
top: Top
, 
prop: ℙ
, 
false: False
, 
rev_implies: P 
⇐ Q
, 
subtype_rel: A ⊆r B
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
int_seg: {i..j-}
, 
sq_type: SQType(T)
, 
guard: {T}
, 
r2-perp: r2-perp(x)
, 
real-vec-mul: a*X
, 
req-vec: req-vec(n;x;y)
, 
eq_int: (i =z j)
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
bfalse: ff
, 
lelt: i ≤ j < k
, 
le: A ≤ B
, 
less_than: a < b
, 
less_than': less_than'(a;b)
, 
true: True
, 
rneq: x ≠ y
, 
real-vec: ℝ^n
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
uiff: uiff(P;Q)
, 
bnot: ¬bb
, 
assert: ↑b
, 
nequal: a ≠ b ∈ T 
, 
rat_term_to_real: rat_term_to_real(f;t)
, 
rtermMultiply: left "*" right
, 
rat_term_ind: rat_term_ind, 
rtermDivide: num "/" denom
, 
rtermMinus: rtermMinus(num)
, 
rtermVar: rtermVar(var)
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
rdiv: (x/y)
, 
req_int_terms: t1 ≡ t2
, 
dot-product: x⋅y
, 
subtract: n - m
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
Lemmas referenced : 
req_wf, 
dot-product_wf, 
nat_plus_properties, 
decidable__le, 
full-omega-unsat, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
istype-int, 
int_formula_prop_not_lemma, 
istype-void, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_wf, 
istype-le, 
int-to-real_wf, 
real_wf, 
req-vec_wf, 
real-vec-mul_wf, 
r2-perp_wf, 
rless_wf, 
real-vec-norm_wf, 
real-vec_wf, 
sq_stable__req, 
real-vec-norm-positive-iff, 
decidable__equal_int, 
subtype_base_sq, 
int_subtype_base, 
int_seg_properties, 
int_seg_subtype_special, 
int_seg_cases, 
intformand_wf, 
intformless_wf, 
itermVar_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
rless-int, 
rless_functionality, 
req_weakening, 
rdiv_wf, 
rminus_wf, 
decidable__lt, 
istype-less_than, 
int_seg_wf, 
eq_int_wf, 
eqtt_to_assert, 
assert_of_eq_int, 
rmul_wf, 
eqff_to_assert, 
bool_cases_sqequal, 
bool_wf, 
bool_subtype_base, 
assert-bnot, 
neg_assert_of_eq_int, 
assert-rat-term-eq2, 
rtermVar_wf, 
rtermMultiply_wf, 
rtermDivide_wf, 
rtermMinus_wf, 
rmul_preserves_req, 
rneq_wf, 
rinv_wf2, 
itermSubtract_wf, 
itermMultiply_wf, 
itermMinus_wf, 
req_functionality, 
req_transitivity, 
rmul_functionality, 
rmul-rinv, 
req-iff-rsub-is-0, 
real_polynomial_null, 
real_term_value_sub_lemma, 
real_term_value_mul_lemma, 
real_term_value_var_lemma, 
real_term_value_const_lemma, 
squash_wf, 
true_wf, 
subtype_rel_self, 
iff_weakening_equal, 
rminus_functionality, 
real_term_value_minus_lemma, 
rsum_wf, 
itermAdd_wf, 
int_term_value_add_lemma, 
radd_wf, 
istype-false, 
intformeq_wf, 
int_formula_prop_eq_lemma, 
rsum-split-first, 
radd_functionality, 
rsum-single, 
real_term_value_add_lemma, 
radd-preserves-req, 
req_inversion, 
req-implies-req, 
rsub_wf, 
dot-product-comm, 
dot-product_functionality, 
req-vec_weakening, 
dot-product-linearity2, 
rmul-zero
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation_alt, 
independent_pairFormation, 
universeIsType, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
dependent_set_memberEquality_alt, 
natural_numberEquality, 
setElimination, 
rename, 
hypothesisEquality, 
hypothesis, 
dependent_functionElimination, 
unionElimination, 
independent_isectElimination, 
approximateComputation, 
independent_functionElimination, 
dependent_pairFormation_alt, 
lambdaEquality_alt, 
isect_memberEquality_alt, 
voidElimination, 
sqequalRule, 
productIsType, 
applyEquality, 
inhabitedIsType, 
equalityTransitivity, 
equalitySymmetry, 
productElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
equalityIstype, 
because_Cache, 
instantiate, 
cumulativity, 
intEquality, 
hypothesis_subsumption, 
int_eqEquality, 
inrFormation_alt, 
inlFormation_alt, 
functionIsType, 
closedConclusion, 
equalityElimination, 
promote_hyp, 
universeEquality, 
addEquality, 
setIsType, 
baseApply, 
sqequalBase
Latex:
\mforall{}x:\mBbbR{}\^{}2.  ((r0  <  ||x||)  {}\mRightarrow{}  (\mforall{}y:\mBbbR{}\^{}2.  (y\mcdot{}x  =  r0  \mLeftarrow{}{}\mRightarrow{}  \mexists{}t:\mBbbR{}.  req-vec(2;y;t*r2-perp(x)))))
Date html generated:
2019_10_30-AM-11_32_23
Last ObjectModification:
2019_04_02-PM-04_21_26
Theory : reals!model!euclidean!geometry
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