Nuprl Lemma : rsqrt-rless-iff
∀x,y:{x:ℝ| r0 ≤ x} . (x < y ⇐⇒ rsqrt(x) < rsqrt(y))
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
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,
le: A ≤ B,
false: False,
not: ¬A,
uimplies: b supposing a
Lemmas referenced :
rsqrt_functionality_wrt_rless,
rless_wf,
rsqrt_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul_wf,
set_wf,
rnexp-rless,
rsqrt_nonneg,
less_than_wf,
rless_functionality,
rnexp_wf,
false_wf,
le_wf,
rsqrt-rnexp-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_functionElimination,
isectElimination,
applyEquality,
lambdaEquality,
setEquality,
productEquality,
natural_numberEquality,
sqequalRule,
because_Cache,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination
Latex:
\mforall{}x,y:\{x:\mBbbR{}| r0 \mleq{} x\} . (x < y \mLeftarrow{}{}\mRightarrow{} rsqrt(x) < rsqrt(y))
Date html generated:
2018_05_22-PM-02_23_13
Last ObjectModification:
2018_03_27-PM-10_22_47
Theory : reals
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