Nuprl Lemma : rv-between-iff
∀n:ℕ. ∀a,b,c:ℝ^n. (a-b-c ⇐⇒ a ≠ b ∧ b ≠ c ∧ a ≠ c ∧ rv-T(n;a;b;c))
Proof
Definitions occuring in Statement :
rv-T: rv-T(n;a;b;c),
rv-between: a-b-c,
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
rv-between: a-b-c,
rv-T: rv-T(n;a;b;c),
not: ¬A,
false: False,
real-vec-be: real-vec-be(n;a;b;c),
exists: ∃x:A. B[x],
real-vec-between: a-b-c,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
req-vec: req-vec(n;x;y),
real-vec-mul: a*X,
real-vec-add: X + Y,
nat: ℕ,
real-vec: ℝ^n,
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
real-vec-sep: a ≠ b,
guard: {T},
true: True,
squash: ↓T,
i-member: r ∈ I,
rccint: [l, u],
rooint: (l, u)
Lemmas referenced :
rv-between_wf,
real-vec-sep_wf,
rv-T_wf,
real-vec_wf,
nat_wf,
rv-between-sep,
rv-between-symmetry,
real-vec-sep-symmetry,
not_wf,
rv-non-strict-between-iff,
i-member_wf,
rooint_wf,
int-to-real_wf,
req-vec_wf,
real-vec-add_wf,
real-vec-mul_wf,
rsub_wf,
real-vec-dist-between-1,
real-vec-dist_wf,
real_wf,
rleq_wf,
rmul_wf,
rabs_wf,
req_functionality,
real-vec-dist_functionality,
req-vec_inversion,
req-vec_weakening,
req_weakening,
int_seg_wf,
radd_wf,
req_wf,
rminus_wf,
uiff_transitivity,
radd_functionality,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
rminus_functionality,
rmul-one-both,
rmul_comm,
rminus-radd,
req_inversion,
radd-assoc,
radd-ac,
radd_comm,
rminus-as-rmul,
rmul_functionality,
rmul-identity1,
rmul-distrib2,
radd-int,
rmul-zero-both,
rminus-rminus,
radd-zero-both,
rmul_preserves_rless,
rless_wf,
rless_functionality,
rless_transitivity1,
rleq_weakening,
radd-preserves-req,
radd-rminus-assoc,
rmul-int,
uiff_transitivity3,
squash_wf,
true_wf,
rminus-int,
rabs_functionality,
real-vec-dist-symmetry,
radd-preserves-rleq,
rabs-of-nonneg,
rleq_functionality,
radd-rminus-both,
radd-preserves-rless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
productEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
voidElimination,
dependent_pairFormation,
natural_numberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
sqequalRule,
independent_isectElimination,
minusEquality,
addEquality,
addLevel,
multiplyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
promote_hyp
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n. (a-b-c \mLeftarrow{}{}\mRightarrow{} a \mneq{} b \mwedge{} b \mneq{} c \mwedge{} a \mneq{} c \mwedge{} rv-T(n;a;b;c))
Date html generated:
2016_10_26-AM-10_46_33
Last ObjectModification:
2016_10_05-PM-01_21_14
Theory : reals
Home
Index