Nuprl Lemma : proj-sep-implies
∀n:ℕ. ∀a,b:ℙ^n. (a ≠ b ⇒ (∃i,j:ℕn + 1. (a i) * (b j) ≠ (a j) * (b i)))
Proof
Definitions occuring in Statement :
proj-sep: a ≠ b,
real-proj: ℙ^n,
rneq: x ≠ y,
rmul: a * b,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
real-proj: ℙ^n,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
proj-sep: a ≠ b,
real-vec-sep: a ≠ b,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
subtype_rel: A ⊆r B,
real: ℝ,
nat_plus: ℕ+,
sq_stable: SqStable(P),
squash: ↓T,
rneq: x ≠ y,
guard: {T},
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
so_lambda: λ2x.t[x],
so_apply: x[s],
punit: u(a),
real-vec-mul: a*X,
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
Cauchy-Schwarz-strict,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
Cauchy-Schwarz-non-equality,
proj-norm-positive,
real-vec-sep-iff-rneq,
sq_stable__less_than,
int-to-real_wf,
real_wf,
real-vec-dist_wf,
nat_plus_properties,
punit_wf,
real-vec-mul_wf,
rdiv_wf,
real-vec-norm_wf,
rless_wf,
rneq_wf,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
real-vec_wf,
rminus_wf,
rneq-symmetry,
rmul_wf,
exists_wf,
int_seg_wf,
proj-sep_wf,
real-proj_wf,
nat_wf,
rmul_preserves_rneq,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
rmul-one,
rneq_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
rmul-rinv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
itermMinus_wf,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
isectElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
productElimination,
because_Cache,
applyEquality,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
inrFormation,
promote_hyp,
addLevel,
levelHypothesis
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbP{}\^{}n. (a \mneq{} b {}\mRightarrow{} (\mexists{}i,j:\mBbbN{}n + 1. (a i) * (b j) \mneq{} (a j) * (b i)))
Date html generated:
2017_10_05-AM-00_19_14
Last ObjectModification:
2017_06_20-PM-02_13_35
Theory : inner!product!spaces
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