Nuprl Lemma : rsqrt-rneq
∀x,y:{x:ℝ| r0 ≤ x} . (rsqrt(x) ≠ rsqrt(y) ⇒ x ≠ y)
Proof
Definitions occuring in Statement :
rsqrt: rsqrt(x),
rneq: x ≠ y,
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rneq: x ≠ y,
or: P ∨ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
guard: {T},
subtype_rel: A ⊆r B,
and: P ∧ Q,
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,
iff: P ⇐⇒ Q
Lemmas referenced :
rless_wf,
rneq_wf,
rsqrt_wf,
rleq_wf,
int-to-real_wf,
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,
sqequalHypSubstitution,
unionElimination,
thin,
inlFormation,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
inrFormation,
dependent_set_memberEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
setEquality,
productEquality,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination
Latex:
\mforall{}x,y:\{x:\mBbbR{}| r0 \mleq{} x\} . (rsqrt(x) \mneq{} rsqrt(y) {}\mRightarrow{} x \mneq{} y)
Date html generated:
2017_10_03-AM-10_44_28
Last ObjectModification:
2017_06_21-PM-11_03_57
Theory : reals
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