Nuprl Lemma : rsqrt-rleq-iff
∀[x:{x:ℝ| r0 ≤ x} ]. ∀[c:ℝ]. uiff(rsqrt(x) ≤ c;(r0 ≤ c) ∧ (x ≤ c^2))
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rleq: x ≤ y,
rnexp: x^k1,
int-to-real: r(n),
real: ℝ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
set: {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,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T}
Lemmas referenced :
le_witness_for_triv,
rleq_wf,
rsqrt_wf,
int-to-real_wf,
rnexp_wf,
istype-void,
istype-le,
real_wf,
rsqrt_nonneg,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rnexp_functionality_wrt_rleq,
rmul_wf,
rleq_functionality,
req_weakening,
rnexp2,
rsqrt_squared,
square-rleq-implies
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality_alt,
dependent_functionElimination,
hypothesisEquality,
extract_by_obid,
isectElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
applyEquality,
setElimination,
rename,
productIsType,
natural_numberEquality,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
voidElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
setIsType,
because_Cache,
independent_functionElimination
Latex:
\mforall{}[x:\{x:\mBbbR{}| r0 \mleq{} x\} ]. \mforall{}[c:\mBbbR{}]. uiff(rsqrt(x) \mleq{} c;(r0 \mleq{} c) \mwedge{} (x \mleq{} c\^{}2))
Date html generated:
2019_10_30-AM-07_57_15
Last ObjectModification:
2019_06_25-PM-03_53_11
Theory : reals
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