Nuprl Lemma : in-compatible-cubes
∀k:ℕ. ∀c,d:ℚCube(k).
(Compatible(c;d)
⇒ (∀p:ℝ^k. (in-rat-cube(k;p;c) ⇒ in-rat-cube(k;p;d) ⇒ (∃f:ℚCube(k). (f ≤ c ∧ f ≤ d ∧ in-rat-cube(k;p;f))))))
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],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
compatible-rat-cubes: Compatible(c;d),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k)
Definitions unfolded in proof :
uimplies: b supposing a,
uiff: uiff(P;Q),
nat: ℕ,
pi2: snd(t),
pi1: fst(t),
rat-interval-intersection: I ⋂ J,
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
real-vec: ℝ^n,
rat-cube-intersection: c ⋂ d,
in-rat-cube: in-rat-cube(k;p;c),
cand: A c∧ B,
prop: ℙ,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
compatible-rat-cubes: Compatible(c;d),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rat2real-qmin,
req_weakening,
rat2real-qmax,
rleq_functionality,
rmin_ub,
rmax_lb,
iff_weakening_uiff,
rmin_wf,
qmin_wf,
qmax_wf,
rmax_wf,
rat2real_wf,
rleq_wf,
int_seg_wf,
istype-nat,
rational-cube_wf,
compatible-rat-cubes_wf,
real-vec_wf,
rat-cube-face_wf,
in-rat-cube_wf,
rat-cube-intersection_wf,
inhabited-iff-in-rat-cube
Rules used in proof :
independent_isectElimination,
promote_hyp,
because_Cache,
productEquality,
rename,
setElimination,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
applyEquality,
inhabitedIsType,
productIsType,
sqequalRule,
independent_pairFormation,
universeIsType,
dependent_pairFormation_alt,
productElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
thin,
independent_functionElimination,
sqequalHypSubstitution,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c,d:\mBbbQ{}Cube(k).
(Compatible(c;d)
{}\mRightarrow{} (\mforall{}p:\mBbbR{}\^{}k
(in-rat-cube(k;p;c)
{}\mRightarrow{} in-rat-cube(k;p;d)
{}\mRightarrow{} (\mexists{}f:\mBbbQ{}Cube(k). (f \mleq{} c \mwedge{} f \mleq{} d \mwedge{} in-rat-cube(k;p;f))))))
Date html generated:
2019_10_31-AM-06_03_32
Last ObjectModification:
2019_10_30-PM-07_01_25
Theory : real!vectors
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