Nuprl Lemma : ball-slice-radius_wf
∀[r:{r:ℝ| r0 ≤ r} ]. ∀[t:{t:ℝ| t ∈ [-(r), r]} ]. (ball-slice-radius(r;t) ∈ {r':ℝ| (r0 ≤ r') ∧ ((r'^2 + t^2) = r^2)} )
Proof
Definitions occuring in Statement :
ball-slice-radius: ball-slice-radius(r;t),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rnexp: x^k1,
req: x = y,
rminus: -(x),
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
ball-slice-radius: ball-slice-radius(r;t),
prop: ℙ,
subtype_rel: A ⊆r B,
cand: A c∧ B,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
i-member: r ∈ I,
rccint: [l, u],
squash: ↓T,
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
radd-preserves-rleq,
int-to-real_wf,
rsub_wf,
rnexp_wf,
istype-void,
istype-le,
rsqrt_wf,
rleq_wf,
rsqrt_nonneg,
req_wf,
radd_wf,
real_wf,
i-member_wf,
rccint_wf,
rminus_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
rabs_wf,
req_weakening,
rnexp_functionality_wrt_rleq,
zero-rleq-rabs,
sq_stable__rleq,
rabs-rleq-iff,
req-iff-rsub-is-0,
rleq_functionality,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
req_functionality,
rabs-rnexp2,
radd_functionality,
rsqrt-rnexp-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
dependent_set_memberEquality_alt,
independent_pairFormation,
sqequalRule,
lambdaFormation_alt,
voidElimination,
hypothesisEquality,
setElimination,
rename,
because_Cache,
productElimination,
independent_isectElimination,
universeIsType,
applyEquality,
productIsType,
setIsType,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt
Latex:
\mforall{}[r:\{r:\mBbbR{}| r0 \mleq{} r\} ]. \mforall{}[t:\{t:\mBbbR{}| t \mmember{} [-(r), r]\} ].
(ball-slice-radius(r;t) \mmember{} \{r':\mBbbR{}| (r0 \mleq{} r') \mwedge{} ((r'\^{}2 + t\^{}2) = r\^{}2)\} )
Date html generated:
2019_10_30-AM-10_14_57
Last ObjectModification:
2019_06_28-PM-01_52_13
Theory : real!vectors
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