Nuprl Lemma : in-rat-cube_functionality
∀[k:ℕ]. ∀[p,q:ℝ^k]. ∀[c:ℚCube(k)]. uiff(in-rat-cube(k;p;c);in-rat-cube(k;q;c)) supposing p ≡ q
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,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
rational-cube: ℚCube(k)
Definitions unfolded in proof :
prop: ℙ,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
nat: ℕ,
pi2: snd(t),
real-vec: ℝ^n,
pi1: fst(t),
rational-interval: ℚInterval,
implies: P ⇒ Q,
rational-cube: ℚCube(k),
guard: {T},
cand: A c∧ B,
req-vec: req-vec(n;x;y),
all: ∀x:A. B[x],
in-rat-cube: in-rat-cube(k;p;c),
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-nat,
rational-cube_wf,
rn-prod-metric_wf,
real-vec_wf,
meq_wf,
in-rat-cube_wf,
le_witness_for_triv,
int_seg_wf,
req_inversion,
rleq_weakening,
rat2real_wf,
rleq_transitivity,
meq-rn-prod-metric
Rules used in proof :
isectIsTypeImplies,
isect_memberEquality_alt,
functionIsTypeImplies,
independent_pairEquality,
lambdaEquality_alt,
rename,
setElimination,
natural_numberEquality,
universeIsType,
because_Cache,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
sqequalRule,
inhabitedIsType,
applyEquality,
dependent_functionElimination,
lambdaFormation_alt,
independent_isectElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
independent_pairFormation,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[p,q:\mBbbR{}\^{}k]. \mforall{}[c:\mBbbQ{}Cube(k)]. uiff(in-rat-cube(k;p;c);in-rat-cube(k;q;c)) supposing p \mequiv{} q
Date html generated:
2019_10_30-AM-10_12_48
Last ObjectModification:
2019_10_29-AM-11_46_37
Theory : real!vectors
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