Nuprl Lemma : rat-point-in-intersection
∀[k:ℕ]. ∀[x:ℕk ⟶ ℚ]. ∀[c,d:ℚCube(k)].
uiff(rat-point-in-cube(k;x;c ⋂ d);rat-point-in-cube(k;x;c) ∧ rat-point-in-cube(k;x;d))
Proof
Definitions occuring in Statement :
rat-point-in-cube: rat-point-in-cube(k;x;c),
rat-cube-intersection: c ⋂ d,
rational-cube: ℚCube(k),
rationals: ℚ,
int_seg: {i..j-},
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
rat-point-in-cube: rat-point-in-cube(k;x;c),
all: ∀x:A. B[x],
rational-cube: ℚCube(k),
implies: P ⇒ Q,
rational-interval: ℚInterval,
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
nat: ℕ,
rat-cube-intersection: c ⋂ d,
rat-interval-intersection: I ⋂ J,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
qle_witness,
rat-point-in-cube_wf,
rat-cube-intersection_wf,
subtype_rel_self,
int_seg_wf,
rational-interval_wf,
rational-cube_wf,
rationals_wf,
istype-nat,
qle_wf,
iff_weakening_uiff,
qmax_wf,
qmax_lb,
qmin_wf,
qmin_ub
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
lambdaFormation_alt,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality_alt,
extract_by_obid,
isectElimination,
applyEquality,
inhabitedIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionIsTypeImplies,
universeIsType,
functionEquality,
setElimination,
rename,
imageElimination,
productIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
natural_numberEquality,
productEquality,
promote_hyp
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[x:\mBbbN{}k {}\mrightarrow{} \mBbbQ{}]. \mforall{}[c,d:\mBbbQ{}Cube(k)].
uiff(rat-point-in-cube(k;x;c \mcap{} d);rat-point-in-cube(k;x;c) \mwedge{} rat-point-in-cube(k;x;d))
Date html generated:
2020_05_20-AM-09_18_12
Last ObjectModification:
2019_11_02-PM-04_37_31
Theory : rationals
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