Nuprl Lemma : extracted-rroot_wf
∀[i:{2...}]. ∀[x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} ].
(extracted-rroot(i;x) ∈ {y:ℝ| ((↑isEven(i)) ⇒ (r0 ≤ y)) ∧ (y^i = x)} )
Proof
Definitions occuring in Statement :
extracted-rroot: extracted-rroot(i;x),
rleq: x ≤ y,
rnexp: x^k1,
req: x = y,
int-to-real: r(n),
real: ℝ,
isEven: isEven(n),
int_upper: {i...},
assert: ↑b,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
uimplies: b supposing a,
sq_exists: ∃x:A [B[x]],
so_apply: x[s],
and: P ∧ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
int_upper: {i...},
implies: P ⇒ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
extracted-rroot: extracted-rroot(i;x),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
set_wf,
false_wf,
upper_subtype_nat,
rnexp_wf,
req_wf,
sq_exists_wf,
int-to-real_wf,
rleq_wf,
isEven_wf,
assert_wf,
real_wf,
all_wf,
int_upper_wf,
subtype_rel_self,
rroot-exists-ext
Rules used in proof :
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
dependent_set_memberEquality,
productEquality,
lambdaEquality,
hypothesisEquality,
because_Cache,
setEquality,
natural_numberEquality,
functionEquality,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
instantiate,
applyEquality,
sqequalRule,
rename,
thin,
setElimination,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[i:\{2...\}]. \mforall{}[x:\{x:\mBbbR{}| (\muparrow{}isEven(i)) {}\mRightarrow{} (r0 \mleq{} x)\} ].
(extracted-rroot(i;x) \mmember{} \{y:\mBbbR{}| ((\muparrow{}isEven(i)) {}\mRightarrow{} (r0 \mleq{} y)) \mwedge{} (y\^{}i = x)\} )
Date html generated:
2018_05_22-PM-02_22_57
Last ObjectModification:
2018_05_21-AM-00_45_41
Theory : reals
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