Nuprl Lemma : rsin-strict-bound
∀x:{x:ℝ| x ∈ (-(π/2), π/2)} . (rsin(x) ∈ (r(-1), r1))
Proof
Definitions occuring in Statement :
halfpi: π/2,
rsin: rsin(x),
rooint: (l, u),
i-member: r ∈ I,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
uimplies: b supposing a,
top: Top,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
rcos-positive,
rsin-rcos-pythag,
set_wf,
real_wf,
i-member_wf,
rooint_wf,
rminus_wf,
halfpi_wf,
radd_wf,
rnexp_wf,
false_wf,
le_wf,
rsin_wf,
rcos_wf,
rmul_wf,
int-to-real_wf,
rmul-is-positive,
rless_wf,
radd-preserves-req,
req_wf,
radd-preserves-rless,
req_functionality,
radd_functionality,
rnexp2,
req_weakening,
uiff_transitivity,
radd_comm,
radd-ac,
radd-rminus-assoc,
rless_functionality,
req_inversion,
radd-assoc,
radd-rminus-both,
radd-zero-both,
square-rless-1-iff,
rabs-rless-iff,
member_rooint_lemma,
squash_wf,
true_wf,
rminus-int,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
isectElimination,
sqequalRule,
lambdaEquality,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
productElimination,
independent_functionElimination,
inlFormation,
productEquality,
independent_isectElimination,
promote_hyp,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}x:\{x:\mBbbR{}| x \mmember{} (-(\mpi{}/2), \mpi{}/2)\} . (rsin(x) \mmember{} (r(-1), r1))
Date html generated:
2016_10_26-PM-00_25_59
Last ObjectModification:
2016_09_12-PM-05_43_50
Theory : reals_2
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