Nuprl Lemma : real-vec-sep-mul
∀n:ℕ. ∀a,b:ℝ. ∀y,y':ℝ^n. (a*y ≠ b*y' ⇒ (a ≠ b ∨ y ≠ y'))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
real-vec-mul: a*X,
real-vec: ℝ^n,
rneq: x ≠ y,
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
real-vec: ℝ^n,
so_apply: x[s],
or: P ∨ Q,
real-vec-mul: a*X,
guard: {T},
uimplies: b supposing a,
rge: x ≥ y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
rneq: x ≠ y
Lemmas referenced :
real-vec-sep-iff,
real-vec-mul_wf,
real-vec-sep_wf,
or_wf,
rneq_wf,
exists_wf,
int_seg_wf,
rless_wf,
int-to-real_wf,
rabs_wf,
rsub_wf,
real-vec_wf,
real_wf,
nat_wf,
r-triangle-inequality2,
rmul_wf,
radd_wf,
radd-positive-implies,
rless_functionality_wrt_implies,
rleq_weakening_equal,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rless_functionality,
req_weakening,
rabs_functionality,
rabs-rmul,
rmul-is-positive,
zero-rleq-rabs,
rless_transitivity1,
rless_irreflexivity,
rneq-if-rabs
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
productElimination,
independent_functionElimination,
addLevel,
orFunctionality,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
unionElimination,
inlFormation,
inrFormation,
dependent_pairFormation,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbR{}. \mforall{}y,y':\mBbbR{}\^{}n. (a*y \mneq{} b*y' {}\mRightarrow{} (a \mneq{} b \mvee{} y \mneq{} y'))
Date html generated:
2017_10_03-AM-11_01_43
Last ObjectModification:
2017_04_07-PM-02_13_58
Theory : reals
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