Nuprl Lemma : rv-sep-between
∀n:ℕ. ∀a,b:ℝ^n. (a ≠ b ⇒ (∃m:ℝ^n. a-m-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],
exists: ∃x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
rv-between: a-b-c,
real-vec-between: a-b-c,
top: Top,
cand: A c∧ B,
nat_plus: ℕ+,
rsub: x - y,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q)
Lemmas referenced :
real-vec-add_wf,
real-vec-mul_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
member_rooint_lemma,
rless-int-fractions2,
less_than_wf,
rless-int-fractions3,
req-vec_wf,
rsub_wf,
rv-between_wf,
real-vec-sep_wf,
real-vec_wf,
nat_wf,
radd-int,
rminus-int,
real_wf,
true_wf,
squash_wf,
uiff_transitivity3,
rmul-rdiv-cancel,
rmul_comm,
rminus_functionality,
rmul-one-both,
rmul_over_rminus,
radd_functionality,
rmul-distrib,
req_transitivity,
rmul-rdiv-cancel2,
req_functionality,
uiff_transitivity,
req_weakening,
rminus_wf,
radd_wf,
rmul_wf,
req_wf,
rmul_preserves_req,
real-vec-mul_functionality,
real-vec-add_functionality,
req-vec_weakening,
req-vec_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
natural_numberEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
multiplyEquality,
productEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
addEquality,
minusEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b:\mBbbR{}\^{}n. (a \mneq{} b {}\mRightarrow{} (\mexists{}m:\mBbbR{}\^{}n. a-m-b))
Date html generated:
2017_10_03-AM-11_14_28
Last ObjectModification:
2017_07_28-AM-08_24_39
Theory : reals
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