Nuprl Lemma : rtan-double
∀[x:{x:ℝ| x ∈ (-(π/2), π/2)} ]. rtan(r(2) * x) = (r(2) * rtan(x)/r1 - rtan(x)^2) supposing r(2) * x ∈ (-(π/2), π/2)
Proof
Definitions occuring in Statement :
rtan: rtan(x),
halfpi: π/2,
rooint: (l, u),
i-member: r ∈ I,
rdiv: (x/y),
rnexp: x^k1,
rsub: x - y,
req: x = y,
rmul: a * b,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
cand: A c∧ B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q
Lemmas referenced :
req_witness,
rtan_wf,
rmul_wf,
int-to-real_wf,
i-member_wf,
rooint_wf,
rminus_wf,
halfpi_wf,
rdiv_wf,
rsub_wf,
rnexp_wf,
false_wf,
le_wf,
rless_wf,
set_wf,
real_wf,
member_rooint_lemma,
rless-implies-rless,
radd_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
itermAdd_wf,
req-iff-rsub-is-0,
rtan-radd-denom-positive,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_add_lemma,
rless_functionality,
req_weakening,
rsub_functionality,
req_inversion,
rnexp2,
rtan-radd,
req_wf,
rless_transitivity1,
rleq_weakening,
rless_transitivity2,
uiff_transitivity,
req_functionality,
rtan_functionality,
rneq_functionality,
rdiv_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality,
natural_numberEquality,
hypothesis,
hypothesisEquality,
because_Cache,
sqequalRule,
independent_pairFormation,
lambdaFormation,
independent_isectElimination,
inrFormation,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
voidElimination,
voidEquality,
productElimination,
approximateComputation,
int_eqEquality,
intEquality,
productEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (-(\mpi{}/2), \mpi{}/2)\} ]
rtan(r(2) * x) = (r(2) * rtan(x)/r1 - rtan(x)\^{}2) supposing r(2) * x \mmember{} (-(\mpi{}/2), \mpi{}/2)
Date html generated:
2018_05_22-PM-03_00_31
Last ObjectModification:
2017_10_19-PM-06_28_02
Theory : reals_2
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