Nuprl Lemma : real-vec-perp-exists
∀n:{2...}. ∀x:ℝ^n. (x ≠ λi.r0 ⇒ (∃y:ℝ^n. (y ≠ λi.r0 ∧ (x⋅y = r0))))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
dot-product: x⋅y,
real-vec: ℝ^n,
req: x = y,
int-to-real: r(n),
int_upper: {i...},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
pointwise-req: x[k] = y[k] for k ∈ [n,m],
rev_uimplies: rev_uimplies(P;Q),
real: ℝ,
so_apply: x[s],
so_lambda: λ2x.t[x],
dot-product: x⋅y,
nequal: a ≠ b ∈ T ,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
squash: ↓T,
sq_stable: SqStable(P),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rev_implies: P ⇐ Q,
cand: A c∧ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
real-vec-sep: a ≠ b,
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
rless: x < y,
lelt: i ≤ j < k,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
decidable: Dec(P),
int_seg: {i..j-},
top: Top,
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
int_upper: {i...},
real-vec: ℝ^n,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
req_inversion,
radd-zero,
radd-preserves-req,
rsum_linearity2,
req_transitivity,
le_wf,
rsum_functionality,
real_term_value_minus_lemma,
itermMinus_wf,
real_term_value_mul_lemma,
real_term_value_add_lemma,
itermMultiply_wf,
radd_functionality,
radd-zero-both,
rmul-zero,
rsum-zero-req,
rsum-split-last,
req_functionality,
int_term_value_add_lemma,
itermAdd_wf,
radd_wf,
rsum_wf,
int_term_value_subtract_lemma,
decidable__le,
real-vec-dist_wf,
sq_stable__less_than,
subtract-add-cancel,
rmul_wf,
subtract_wf,
rsum-split,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
rless_wf,
dot-product_wf,
req_wf,
rminus_wf,
real_wf,
eq_int_wf,
ifthenelse_wf,
equal_wf,
equal-wf-base-T,
equal-wf-T-base,
not_wf,
equal-wf-base,
int_formula_prop_eq_lemma,
intformeq_wf,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
int_upper_properties,
int_seg_properties,
nat_plus_properties,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rabs_functionality,
req_weakening,
rless_functionality,
req-iff-rsub-is-0,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
rsub_wf,
rabs_wf,
int_upper_wf,
real-vec_wf,
int_seg_wf,
int-to-real_wf,
false_wf,
upper_subtype_nat,
real-vec-sep_wf,
real-vec-sep-iff
Rules used in proof :
minusEquality,
functionExtensionality,
addEquality,
promote_hyp,
imageElimination,
imageMemberEquality,
equalityElimination,
productEquality,
baseClosed,
dependent_set_memberEquality,
dependent_pairFormation,
cumulativity,
instantiate,
unionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
equalitySymmetry,
equalityTransitivity,
int_eqEquality,
approximateComputation,
rename,
setElimination,
lambdaEquality,
independent_pairFormation,
independent_isectElimination,
natural_numberEquality,
applyEquality,
isectElimination,
sqequalRule,
hypothesis,
independent_functionElimination,
productElimination,
hypothesisEquality,
because_Cache,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n:\{2...\}. \mforall{}x:\mBbbR{}\^{}n. (x \mneq{} \mlambda{}i.r0 {}\mRightarrow{} (\mexists{}y:\mBbbR{}\^{}n. (y \mneq{} \mlambda{}i.r0 \mwedge{} (x\mcdot{}y = r0))))
Date html generated:
2018_05_22-PM-02_27_07
Last ObjectModification:
2018_05_21-AM-01_02_26
Theory : reals
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