Nuprl Lemma : rv-non-strict-between-iff
∀n:ℕ. ∀a,b,c:ℝ^n. (a ≠ c ⇒ (¬(a ≠ b ∧ b ≠ c ∧ (¬a-b-c)) ⇐⇒ real-vec-be(n;a;b;c)))
Proof
Definitions occuring in Statement :
rv-between: a-b-c,
real-vec-sep: a ≠ b,
real-vec-be: real-vec-be(n;a;b;c),
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
stable: Stable{P},
uimplies: b supposing a,
or: P ∨ Q,
rv-between: a-b-c,
real-vec-between: a-b-c,
exists: ∃x:A. B[x],
real-vec-be: real-vec-be(n;a;b;c),
cand: A c∧ B,
top: Top,
guard: {T},
i-member: r ∈ I,
rccint: [l, u],
le: A ≤ B,
less_than': less_than'(a;b),
real-vec-mul: a*X,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
nat: ℕ,
real-vec: ℝ^n,
uiff: uiff(P;Q),
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q),
rooint: (l, u)
Lemmas referenced :
not_wf,
real-vec-sep_wf,
rv-between_wf,
real-vec-be_wf,
real-vec_wf,
nat_wf,
stable_real-vec-be,
false_wf,
or_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
member_rooint_lemma,
member_rccint_lemma,
rleq_weakening_rless,
int-to-real_wf,
i-member_wf,
rccint_wf,
req-vec_wf,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
rleq_weakening_equal,
rleq-int,
int_seg_wf,
req_wf,
radd_wf,
rmul_wf,
subtract_wf,
req_weakening,
not-real-vec-sep-iff-eq,
uiff_transitivity,
req_functionality,
radd_functionality,
rmul_functionality,
rsub-int,
rmul-zero-both,
rmul-one-both,
radd-zero-both,
req-vec_functionality,
req-vec_weakening,
radd_comm,
req-vec_inversion,
rless_wf,
rooint_wf,
not-rless,
rleq_antisymmetry,
real-vec-add_functionality,
real-vec-mul_functionality,
rsub_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesisEquality,
hypothesis,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
independent_isectElimination,
functionEquality,
unionElimination,
productElimination,
dependent_pairFormation,
isect_memberEquality,
voidEquality,
natural_numberEquality,
sqequalRule,
because_Cache,
setElimination,
rename,
applyEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n. (a \mneq{} c {}\mRightarrow{} (\mneg{}(a \mneq{} b \mwedge{} b \mneq{} c \mwedge{} (\mneg{}a-b-c)) \mLeftarrow{}{}\mRightarrow{} real-vec-be(n;a;b;c)))
Date html generated:
2016_10_26-AM-10_45_06
Last ObjectModification:
2016_09_29-PM-09_07_20
Theory : reals
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