Nuprl Lemma : real-cube-sep-disjoint
∀[k:ℕ]. ∀[c1,c2:real-cube(k)]. (c1 # c2 ⇒ (∀p:ℝ^k. (¬(p ∈ c1 ∧ p ∈ c2))))
Proof
Definitions occuring in Statement :
real-cube-sep: c1 # c2,
in-real-cube: p ∈ c,
real-cube: real-cube(k),
real-vec: ℝ^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
false: False,
and: P ∧ Q,
real-cube-sep: c1 # c2,
exists: ∃x:A. B[x],
in-real-cube: p ∈ c,
or: P ∨ Q,
guard: {T},
subtype_rel: A ⊆r B,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
prop: ℙ
Lemmas referenced :
rless_transitivity1,
cube-upper_wf,
subtype_rel_self,
int_seg_wf,
real_wf,
cube-lower_wf,
rless_irreflexivity,
in-real-cube_wf,
real-vec_wf,
real-cube-sep_wf,
real-cube_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
thin,
sqequalHypSubstitution,
productElimination,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
hypothesis,
extract_by_obid,
applyEquality,
isectElimination,
sqequalRule,
functionEquality,
setElimination,
rename,
imageElimination,
independent_functionElimination,
independent_isectElimination,
because_Cache,
voidElimination,
productIsType,
universeIsType,
lambdaEquality_alt,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c1,c2:real-cube(k)]. (c1 \# c2 {}\mRightarrow{} (\mforall{}p:\mBbbR{}\^{}k. (\mneg{}(p \mmember{} c1 \mwedge{} p \mmember{} c2))))
Date html generated:
2019_10_30-AM-11_31_29
Last ObjectModification:
2019_09_27-PM-01_30_54
Theory : real!vectors
Home
Index