Nuprl Lemma : in-rat-cube-intersection
∀k:ℕ. ∀c,d:ℚCube(k). ∀x:ℝ^k. (in-rat-cube(k;x;c ⋂ d) ⇐⇒ in-rat-cube(k;x;c) ∧ in-rat-cube(k;x;d))
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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rat-cube-intersection: c ⋂ d,
rational-cube: ℚCube(k)
Definitions unfolded in proof :
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
or: P ∨ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
nat: ℕ,
prop: ℙ,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
pi2: snd(t),
pi1: fst(t),
rat-interval-intersection: I ⋂ J,
rational-interval: ℚInterval,
real-vec: ℝ^n,
rational-cube: ℚCube(k),
rat-cube-intersection: c ⋂ d,
member: t ∈ T,
in-rat-cube: in-rat-cube(k;p;c),
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rat2real-qmin,
rmin_ub,
rmin_wf,
req_weakening,
rat2real-qmax,
rleq_functionality,
rmax_lb,
rmax_wf,
qmin_lb,
qmax_ub,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
qle_wf,
qle_reflexivity,
rleq-rat2real,
istype-nat,
rational-cube_wf,
real-vec_wf,
rat-cube-intersection_wf,
in-rat-cube_wf,
int_seg_wf,
qmin_wf,
qmax_wf,
rat2real_wf,
rleq_wf
Rules used in proof :
inrFormation_alt,
inlFormation_alt,
independent_isectElimination,
because_Cache,
rename,
setElimination,
natural_numberEquality,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
isectElimination,
extract_by_obid,
introduction,
universeIsType,
productIsType,
productElimination,
inhabitedIsType,
applyEquality,
sqequalRule,
hypothesisEquality,
thin,
dependent_functionElimination,
hypothesis,
cut,
sqequalHypSubstitution,
independent_pairFormation,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c,d:\mBbbQ{}Cube(k). \mforall{}x:\mBbbR{}\^{}k. (in-rat-cube(k;x;c \mcap{} d) \mLeftarrow{}{}\mRightarrow{} in-rat-cube(k;x;c) \mwedge{} in-rat-cube(k;x;d))
Date html generated:
2019_11_04-PM-04_43_06
Last ObjectModification:
2019_11_04-PM-03_55_44
Theory : real!vectors
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