Nuprl Lemma : deriviative-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 :
member: t ∈ T,
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
rfun-eq: rfun-eq(I;f;g),
all: ∀x:A. B[x],
r-ap: f(x),
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
derivative-cosine,
riiint_wf,
cosine_wf,
real_wf,
i-member_wf,
rcos_wf,
rminus_wf,
sine_wf,
rsin_wf,
req_weakening,
set_wf,
derivative_functionality,
req_functionality,
rcos-is-cosine,
rminus_functionality,
rsin-is-sine
Rules used in proof :
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
lambdaEquality,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
setEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
lambdaFormation,
productElimination
Latex:
d(rcos(x))/dx = \mlambda{}x.-(rsin(x)) on (-\minfty{}, \minfty{})
Date html generated:
2016_10_26-PM-00_14_45
Last ObjectModification:
2016_09_12-PM-05_40_32
Theory : reals_2
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