Nuprl Lemma : rsqrt-rmul
∀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),
rleq: x ≤ y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
uimplies: b supposing a,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rmul_wf,
rsqrt_wf,
rleq_wf,
int-to-real_wf,
real_wf,
req_wf,
rmul-nonneg-case1,
sq_stable__rleq,
set_wf,
rsqrt-unique,
rsqrt_nonneg,
req_functionality,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul_functionality,
req_weakening,
rsqrt_squared,
rmul-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
natural_numberEquality,
applyEquality,
lambdaEquality,
setEquality,
productEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination
Latex:
\mforall{}x:\{x:\mBbbR{}| r0 \mleq{} x\} . \mforall{}[y:\{x:\mBbbR{}| r0 \mleq{} x\} ]. ((rsqrt(x) * rsqrt(y)) = rsqrt(x * y))
Date html generated:
2017_10_03-AM-10_43_55
Last ObjectModification:
2017_07_28-AM-08_18_45
Theory : reals
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