Nuprl Lemma : derivative-rcos
d(rcos(x))/dx = λx.-(rsin(x)) on (-∞, ∞)
Proof
Definitions occuring in Statement :
rcos: rcos(x),
rsin: rsin(x),
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
rminus: -(x)
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
r-ap: f(x),
all: ∀x:A. B[x],
rfun-eq: rfun-eq(I;f;g),
implies: P ⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
uall: ∀[x:A]. B[x],
rfun: I ⟶ℝ,
member: t ∈ T
Lemmas referenced :
rsin-is-sine,
rminus_functionality,
rcos-is-cosine,
req_functionality,
derivative_functionality,
set_wf,
req_weakening,
rsin_wf,
sine_wf,
rminus_wf,
rcos_wf,
i-member_wf,
real_wf,
cosine_wf,
riiint_wf,
derivative-cosine
Rules used in proof :
productElimination,
lambdaFormation,
independent_functionElimination,
independent_isectElimination,
because_Cache,
setEquality,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
lambdaEquality,
sqequalRule,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
hypothesis,
extract_by_obid,
introduction,
cut
Latex:
d(rcos(x))/dx = \mlambda{}x.-(rsin(x)) on (-\minfty{}, \minfty{})
Date html generated:
2018_05_22-PM-02_58_40
Last ObjectModification:
2017_10_20-PM-00_05_43
Theory : reals_2
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