Nuprl Lemma : rat-point-in-cube-interior-not-in-face
∀[k:ℕ]. ∀[x:ℕk ⟶ ℚ]. ∀[a:ℚCube(k)].
∀f:ℚCube(k). (f ≤ a ⇒ rat-point-in-cube(k;x;f) ⇒ (f = a ∈ ℚCube(k))) supposing rat-point-in-cube-interior(k;x;a)
Proof
Definitions occuring in Statement :
rat-point-in-cube-interior: rat-point-in-cube-interior(k;x;a),
rat-point-in-cube: rat-point-in-cube(k;x;c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
rationals: ℚ,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rational-cube: ℚCube(k),
rat-point-in-cube-interior: rat-point-in-cube-interior(k;x;a),
rat-cube-face: c ≤ d,
rat-point-in-cube: rat-point-in-cube(k;x;c),
rational-interval: ℚInterval,
rat-interval-face: I ≤ J,
pi1: fst(t),
pi2: snd(t),
rat-point-interval: [a],
and: P ∧ Q,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
prop: ℙ,
decidable: Dec(P),
guard: {T},
false: False,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
nat: ℕ
Lemmas referenced :
pi2_wf,
rationals_wf,
pi1_wf_top,
istype-void,
qle_wf,
decidable__qless,
qless_transitivity_2_qorder,
qless_irreflexivity,
qless_complement_qorder,
qle_antisymmetry,
qle_transitivity_qorder,
equal_wf,
rational-interval_wf,
qless_wf,
rat-point-in-cube_wf,
rat-cube-face_wf,
rat-point-in-cube-interior_wf,
rational-cube_wf,
int_seg_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
functionExtensionality,
rename,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
inhabitedIsType,
hypothesis,
productElimination,
sqequalRule,
unionElimination,
applyLambdaEquality,
extract_by_obid,
isectElimination,
lambdaEquality_alt,
independent_pairEquality,
isect_memberEquality_alt,
voidElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
independent_functionElimination,
because_Cache,
independent_isectElimination,
productIsType,
universeIsType,
functionIsType,
unionIsType,
equalityIstype,
equalityTransitivity,
axiomEquality,
functionIsTypeImplies,
isectIsTypeImplies,
natural_numberEquality,
setElimination
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[x:\mBbbN{}k {}\mrightarrow{} \mBbbQ{}]. \mforall{}[a:\mBbbQ{}Cube(k)].
\mforall{}f:\mBbbQ{}Cube(k). (f \mleq{} a {}\mRightarrow{} rat-point-in-cube(k;x;f) {}\mRightarrow{} (f = a))
supposing rat-point-in-cube-interior(k;x;a)
Date html generated:
2020_05_20-AM-09_19_01
Last ObjectModification:
2019_11_02-PM-07_42_01
Theory : rationals
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