Nuprl Lemma : meq-in-0-dim-cube
∀[k:ℕ]. ∀[c:ℚCube(k)]. ∀[p,q:ℝ^k]. (p ≡ q) supposing (in-rat-cube(k;q;c) and in-rat-cube(k;p;c)) supposing dim(c) = 0 \000C∈ ℤ
Proof
Definitions occuring in Statement :
in-rat-cube: in-rat-cube(k;p;c),
rn-prod-metric: rn-prod-metric(n),
real-vec: ℝ^n,
meq: x ≡ y,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k)
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
pi1: fst(t),
rational-interval: ℚInterval,
all: ∀x:A. B[x],
rational-cube: ℚCube(k),
real-vec: ℝ^n,
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
implies: P ⇒ Q,
metric: metric(X),
subtype_rel: A ⊆r B,
meq: x ≡ y,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
req-vec_weakening,
req-vec_functionality,
meq-rn-prod-metric,
in-rat-cube_wf,
iff_weakening_uiff,
real-vec_wf,
int_seg_wf,
rat2real_wf,
req-vec_wf,
istype-nat,
rational-cube_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
istype-int,
int-to-real_wf,
rn-prod-metric_wf,
req_witness,
in-0-dim-cube
Rules used in proof :
because_Cache,
dependent_functionElimination,
productElimination,
lambdaFormation_alt,
universeIsType,
sqequalBase,
baseClosed,
addEquality,
minusEquality,
intEquality,
equalityIstype,
isectIsTypeImplies,
independent_functionElimination,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
rename,
setElimination,
lambdaEquality_alt,
applyEquality,
isect_memberEquality_alt,
sqequalRule,
independent_isectElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c:\mBbbQ{}Cube(k)].
\mforall{}[p,q:\mBbbR{}\^{}k]. (p \mequiv{} q) supposing (in-rat-cube(k;q;c) and in-rat-cube(k;p;c)) supposing dim(c) = 0
Date html generated:
2019_10_30-AM-10_12_56
Last ObjectModification:
2019_10_29-PM-03_03_59
Theory : real!vectors
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