Nuprl Lemma : inhabited-rat-cube-iff-point
∀[k:ℕ]. ∀c:ℚCube(k). uiff(↑Inhabited(c);∃x:ℕk ⟶ ℚ. rat-point-in-cube(k;x;c))
Proof
Definitions occuring in Statement :
inhabited-rat-cube: Inhabited(c),
rat-point-in-cube: rat-point-in-cube(k;x;c),
rational-cube: ℚCube(k),
rationals: ℚ,
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
rat-point-in-cube: rat-point-in-cube(k;x;c),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
rational-cube: ℚCube(k),
implies: P ⇒ Q,
nat: ℕ,
exists: ∃x:A. B[x],
rational-interval: ℚInterval,
pi1: fst(t),
prop: ℙ,
pi2: snd(t),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
inhabited-rat-interval: Inhabited(I),
cand: A c∧ B,
guard: {T}
Lemmas referenced :
assert_witness,
inhabited-rat-interval_wf,
int_seg_wf,
istype-assert,
rationals_wf,
qle_wf,
rational-cube_wf,
istype-nat,
inhabited-rat-cube_wf,
iff_weakening_uiff,
assert_wf,
assert-inhabited-rat-cube,
rat-point-in-cube_wf,
qle_reflexivity,
qle_witness,
assert-q_le-eq,
iff_weakening_equal,
q_le_wf,
qle_transitivity_qorder
Rules used in proof :
cut,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
introduction,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
because_Cache,
hypothesis,
independent_functionElimination,
functionIsTypeImplies,
inhabitedIsType,
rename,
functionIsType,
universeIsType,
natural_numberEquality,
setElimination,
productElimination,
productIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionEquality,
promote_hyp,
dependent_pairFormation_alt,
independent_pairEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}c:\mBbbQ{}Cube(k). uiff(\muparrow{}Inhabited(c);\mexists{}x:\mBbbN{}k {}\mrightarrow{} \mBbbQ{}. rat-point-in-cube(k;x;c))
Date html generated:
2020_05_20-AM-09_18_25
Last ObjectModification:
2019_11_02-PM-04_30_23
Theory : rationals
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