Nuprl Lemma : rsin-positive
∀x:{x:ℝ| x ∈ (r0, π)} . (r0 < rsin(x))
Proof
Definitions occuring in Statement :
pi: π,
rsin: rsin(x),
rooint: (l, u),
i-member: r ∈ I,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
i-member: r ∈ I,
rooint: (l, u),
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
uiff: uiff(P;Q),
pi: π,
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y
Lemmas referenced :
rcos-positive,
member_rooint_lemma,
set_wf,
real_wf,
i-member_wf,
rooint_wf,
int-to-real_wf,
pi_wf,
rsub_wf,
halfpi_wf,
radd-preserves-rless,
rminus_wf,
rless_wf,
radd_wf,
rmul_wf,
rless_functionality,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermMinus_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
itermMultiply_wf,
real_term_value_mul_lemma,
int-rmul_wf,
req_weakening,
int-rmul-req,
rsin_functionality,
radd-rminus-assoc,
radd_comm,
radd_functionality,
radd-ac,
req_transitivity,
radd-assoc,
req_inversion,
req_functionality,
uiff_transitivity,
rsin_wf,
req_wf,
rsin-shift-half-pi,
rcos_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
productEquality,
independent_isectElimination,
computeAll,
int_eqEquality,
intEquality,
addLevel,
levelHypothesis,
promote_hyp,
andLevelFunctionality
Latex:
\mforall{}x:\{x:\mBbbR{}| x \mmember{} (r0, \mpi{})\} . (r0 < rsin(x))
Date html generated:
2017_10_04-PM-10_25_44
Last ObjectModification:
2017_07_28-AM-08_49_31
Theory : reals_2
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