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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