Nuprl Lemma : rcos-nonzero-on
rcos(x)≠r0 for x ∈ (-(π/2), π/2)
Proof
Definitions occuring in Statement :
halfpi: π/2,
rcos: rcos(x),
nonzero-on: f[x]≠r0 for x ∈ I,
rooint: (l, u),
rminus: -(x)
Definitions unfolded in proof :
guard: {T},
cand: A c∧ B,
or: P ∨ Q,
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
so_apply: x[s],
prop: ℙ,
rfun: I ⟶ℝ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
rleq_weakening_rless,
radd-zero,
radd-rminus,
radd_wf,
rleq_functionality,
radd-preserves-rleq,
rless_wf,
int-to-real_wf,
rleq_wf,
all_wf,
rcos-positive,
derivative-rsin,
derivative-minus,
derivative_functionality2,
subinterval-riiint,
riiint_wf,
derivative-rcos,
set_wf,
req_wf,
req_weakening,
rcos_functionality,
rminus_functionality,
req_functionality,
rsin_wf,
i-member_wf,
real_wf,
rcos_wf,
halfpi-interval-proper,
halfpi_wf,
rminus_wf,
rooint_wf,
second-deriv-implies-nonzero-on
Rules used in proof :
functionEquality,
natural_numberEquality,
productEquality,
dependent_set_memberEquality,
independent_pairFormation,
inlFormation,
productElimination,
independent_isectElimination,
lambdaFormation,
because_Cache,
setEquality,
hypothesisEquality,
rename,
setElimination,
lambdaEquality,
sqequalRule,
independent_functionElimination,
hypothesis,
isectElimination,
thin,
dependent_functionElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut
Latex:
rcos(x)\mneq{}r0 for x \mmember{} (-(\mpi{}/2), \mpi{}/2)
Date html generated:
2018_05_22-PM-02_59_08
Last ObjectModification:
2018_05_20-PM-11_08_09
Theory : reals_2
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