Nuprl Lemma : rsqrt-rdiv
∀x:{x:ℝ| r0 ≤ x} . ∀y:{x:ℝ| r0 < x} . ((rsqrt(x)/rsqrt(y)) = rsqrt((x/y)))
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
set_wf,
real_wf,
rless_wf,
int-to-real_wf,
rleq_wf,
rsqrt-unique,
rdiv_wf,
sq_stable__rless,
rmul_preserves_rleq,
rmul_wf,
sq_stable__rleq,
uiff_transitivity,
rleq_functionality,
req_weakening,
rmul-rdiv-cancel2,
rmul-zero-both,
rsqrt_wf,
subtype_rel_sets,
rleq_weakening_rless,
rsqrt-positive,
req_wf,
rsqrt_nonneg,
rnexp_wf,
false_wf,
le_wf,
req_functionality,
req_inversion,
rnexp2,
rnexp-rdiv,
rneq_functionality,
rsqrt-rnexp-2,
rdiv_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
independent_isectElimination,
dependent_set_memberEquality,
setElimination,
rename,
because_Cache,
inrFormation,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
applyEquality,
setEquality,
productEquality,
independent_pairFormation
Latex:
\mforall{}x:\{x:\mBbbR{}| r0 \mleq{} x\} . \mforall{}y:\{x:\mBbbR{}| r0 < x\} . ((rsqrt(x)/rsqrt(y)) = rsqrt((x/y)))
Date html generated:
2016_10_26-AM-10_10_05
Last ObjectModification:
2016_10_12-PM-09_07_46
Theory : reals
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