Nuprl Lemma : derivative-rtan
d(rtan(x))/dx = λx.(r1/rcos(x)^2) on (-(π/2), π/2)
Proof
Definitions occuring in Statement :
rtan: rtan(x),
halfpi: π/2,
rcos: rcos(x),
derivative: d(f[x])/dx = λz.g[z] on I,
rooint: (l, u),
rdiv: (x/y),
rnexp: x^k1,
rminus: -(x),
int-to-real: r(n),
natural_number: $n
Definitions unfolded in proof :
top: Top,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
guard: {T},
rneq: x ≠ y,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
cand: A c∧ B,
or: P ∨ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rtan: rtan(x),
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
implies: P ⇒ Q,
all: ∀x:A. B[x],
prop: ℙ,
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Lemmas referenced :
radd_comm,
rsin-rcos-pythag,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-rinv,
rmul_functionality,
rmul-rinv3,
radd_functionality,
req_transitivity,
rsub_functionality,
rdiv_functionality,
itermConstant_wf,
req-iff-rsub-is-0,
itermAdd_wf,
itermMinus_wf,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
radd_wf,
rmul_preserves_req,
rnexp2,
req_inversion,
rless_functionality,
derivative_functionality,
le_wf,
false_wf,
rnexp_wf,
int-to-real_wf,
rless_wf,
rmul-is-positive,
rmul_wf,
rsub_wf,
rdiv_wf,
rtan_wf,
derivative-rcos,
derivative_functionality2,
subinterval-riiint,
riiint_wf,
derivative-rsin,
derivative-rdiv,
rcos-nonzero-on,
rsin_functionality,
rminus_functionality,
set_wf,
req_wf,
req_weakening,
rcos_functionality,
req_functionality,
rcos_wf,
i-member_wf,
real_wf,
rsin_wf,
halfpi_wf,
rminus_wf,
rooint_wf,
rcos-positive
Rules used in proof :
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
inrFormation,
dependent_set_memberEquality,
natural_numberEquality,
productEquality,
independent_pairFormation,
inlFormation,
independent_functionElimination,
dependent_functionElimination,
productElimination,
independent_isectElimination,
lambdaFormation,
because_Cache,
setEquality,
hypothesisEquality,
rename,
setElimination,
lambdaEquality,
sqequalRule,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
d(rtan(x))/dx = \mlambda{}x.(r1/rcos(x)\^{}2) on (-(\mpi{}/2), \mpi{}/2)
Date html generated:
2018_05_22-PM-02_59_30
Last ObjectModification:
2018_05_20-PM-11_09_13
Theory : reals_2
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