Nuprl Lemma : rv-between-sep
∀n:ℕ. ∀a,b,c:ℝ^n. (a-b-c ⇒ a ≠ b)
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rv-between: a-b-c,
and: P ∧ Q,
real-vec-between: a-b-c,
exists: ∃x:A. B[x],
member: t ∈ T,
top: Top,
real-vec-sep: a ≠ b,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rsub: x - y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
guard: {T}
Lemmas referenced :
member_rooint_lemma,
rv-between_wf,
real-vec_wf,
nat_wf,
int-to-real_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
rless_functionality,
req_weakening,
real-vec-dist_functionality,
req-vec_weakening,
rmul_wf,
rabs_wf,
real-vec-dist-between-1,
rmul_preserves_rless,
radd_wf,
rminus_wf,
rmul-zero-both,
rmul_comm,
radd-preserves-rleq,
rleq_weakening_rless,
radd-preserves-rless,
rless_wf,
rabs-of-nonneg,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd-ac,
radd_functionality,
radd-rminus-both,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
hypothesisEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
promote_hyp,
addLevel,
levelHypothesis
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n. (a-b-c {}\mRightarrow{} a \mneq{} b)
Date html generated:
2016_10_26-AM-10_37_28
Last ObjectModification:
2016_09_25-AM-00_09_42
Theory : reals
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