Nuprl Lemma : inhabited-iff-in-rat-cube
∀[k:ℕ]. ∀c:ℚCube(k). (↑Inhabited(c) 
⇐⇒ ∃p:ℝ^k. in-rat-cube(k;p;c))
Proof
Definitions occuring in Statement : 
in-rat-cube: in-rat-cube(k;p;c)
, 
real-vec: ℝ^n
, 
nat: ℕ
, 
assert: ↑b
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
inhabited-rat-cube: Inhabited(c)
, 
rational-cube: ℚCube(k)
Definitions unfolded in proof : 
guard: {T}
, 
le: A ≤ B
, 
rnonneg: rnonneg(x)
, 
rleq: x ≤ y
, 
cand: A c∧ B
, 
pi2: snd(t)
, 
inhabited-rat-interval: Inhabited(I)
, 
in-rat-cube: in-rat-cube(k;p;c)
, 
nat: ℕ
, 
pi1: fst(t)
, 
rational-interval: ℚInterval
, 
rational-cube: ℚCube(k)
, 
real-vec: ℝ^n
, 
prop: ℙ
, 
exists: ∃x:A. B[x]
, 
rev_uimplies: rev_uimplies(P;Q)
, 
rev_implies: P 
⇐ Q
, 
uimplies: b supposing a
, 
uiff: uiff(P;Q)
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
iff: P 
⇐⇒ Q
, 
all: ∀x:A. B[x]
, 
uall: ∀[x:A]. B[x]
Lemmas referenced : 
rleq_wf, 
rleq_transitivity, 
q_le_wf, 
iff_weakening_equal, 
assert-q_le-eq, 
qle_wf, 
le_witness_for_triv, 
rleq-rat2real, 
rleq_weakening_equal, 
int_seg_wf, 
rat2real_wf, 
istype-nat, 
rational-cube_wf, 
in-rat-cube_wf, 
real-vec_wf, 
assert_witness, 
inhabited-rat-cube_wf, 
istype-assert, 
assert-inhabited-rat-cube
Rules used in proof : 
functionIsTypeImplies, 
independent_pairEquality, 
rename, 
setElimination, 
natural_numberEquality, 
dependent_functionElimination, 
equalitySymmetry, 
equalityTransitivity, 
equalityIstype, 
inhabitedIsType, 
applyEquality, 
lambdaEquality_alt, 
dependent_pairFormation_alt, 
universeIsType, 
productIsType, 
sqequalRule, 
independent_functionElimination, 
because_Cache, 
independent_isectElimination, 
productElimination, 
hypothesis, 
hypothesisEquality, 
thin, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
independent_pairFormation, 
lambdaFormation_alt, 
isect_memberFormation_alt, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}].  \mforall{}c:\mBbbQ{}Cube(k).  (\muparrow{}Inhabited(c)  \mLeftarrow{}{}\mRightarrow{}  \mexists{}p:\mBbbR{}\^{}k.  in-rat-cube(k;p;c))
Date html generated:
2019_10_30-AM-10_13_02
Last ObjectModification:
2019_10_29-PM-01_49_09
Theory : real!vectors
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