Nuprl Lemma : rsqrt_functionality_wrt_rleq
∀[x:{x:ℝ| r0 ≤ x} ]. ∀[y:ℝ]. rsqrt(x) ≤ rsqrt(y) supposing x ≤ y
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
uiff: uiff(P;Q),
not: ¬A,
implies: P ⇒ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
nat: ℕ,
iff: P ⇐⇒ Q
Lemmas referenced :
rleq-iff-not-rless,
rsqrt_wf,
rleq_transitivity,
int-to-real_wf,
rleq_wf,
real_wf,
req_wf,
rmul_wf,
rless_wf,
less_than'_wf,
rsub_wf,
nat_plus_wf,
set_wf,
rnexp-rless,
rsqrt_nonneg,
less_than_wf,
rless_functionality,
rnexp_wf,
false_wf,
le_wf,
rsqrt-rnexp-2,
rless_transitivity1,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
natural_numberEquality,
independent_isectElimination,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
setEquality,
productEquality,
productElimination,
lambdaFormation,
dependent_functionElimination,
independent_pairEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[x:\{x:\mBbbR{}| r0 \mleq{} x\} ]. \mforall{}[y:\mBbbR{}]. rsqrt(x) \mleq{} rsqrt(y) supposing x \mleq{} y
Date html generated:
2016_10_26-AM-10_10_18
Last ObjectModification:
2016_09_06-PM-01_21_58
Theory : reals
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