Nuprl Lemma : intersecting-0-dim-cubes
∀k:ℕ. ∀b:ℚCube(k). ∀q:ℝ^k. ∀c:ℚCube(k).
((in-rat-cube(k;q;c) ∧ in-rat-cube(k;q;b) ∧ (dim(c) = 0 ∈ ℤ) ∧ (dim(b) = 0 ∈ ℤ)) ⇒ (c = b ∈ ℚCube(k)))
Proof
Definitions occuring in Statement :
in-rat-cube: in-rat-cube(k;p;c),
real-vec: ℝ^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k)
Definitions unfolded in proof :
pi2: snd(t),
so_apply: x[s],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
subtype_rel: A ⊆r B,
prop: ℙ,
iff: P ⇐⇒ Q,
req-vec: req-vec(n;x;y),
nat: ℕ,
pi1: fst(t),
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
real-vec: ℝ^n,
guard: {T},
uiff: uiff(P;Q),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
istype-nat,
rational-cube_wf,
real-vec_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
istype-int,
in-rat-cube_wf,
rat-cube-dimension-zero,
req-rat2real,
rationals_wf,
equal_wf,
req_wf,
iff_weakening_uiff,
req-vec_transitivity,
int_seg_wf,
rat2real_wf,
req-vec_inversion,
in-0-dim-cube
Rules used in proof :
independent_pairEquality,
sqequalBase,
baseClosed,
addEquality,
minusEquality,
intEquality,
productIsType,
functionExtensionality,
because_Cache,
rename,
setElimination,
natural_numberEquality,
universeIsType,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
inhabitedIsType,
applyEquality,
lambdaEquality_alt,
sqequalRule,
hypothesis,
independent_isectElimination,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
thin,
productElimination,
sqequalHypSubstitution,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}b:\mBbbQ{}Cube(k). \mforall{}q:\mBbbR{}\^{}k. \mforall{}c:\mBbbQ{}Cube(k).
((in-rat-cube(k;q;c) \mwedge{} in-rat-cube(k;q;b) \mwedge{} (dim(c) = 0) \mwedge{} (dim(b) = 0)) {}\mRightarrow{} (c = b))
Date html generated:
2019_10_30-AM-10_13_00
Last ObjectModification:
2019_10_29-PM-04_27_34
Theory : real!vectors
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