Nuprl Lemma : arctangent-negative-rinv
∀[x:{x:ℝ| x ∈ (-∞, r0)} ]. (arctangent(rinv(x)) = (-(π/2) - arctangent(x)))
Proof
Definitions occuring in Statement :
arctangent: arctangent(x),
halfpi: π/2,
rioint: (-∞, u),
i-member: r ∈ I,
rsub: x - y,
rinv: rinv(x),
req: x = y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
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,
all: ∀x:A. B[x],
top: Top,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
rneq: x ≠ y,
or: P ∨ Q,
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
and: P ∧ Q,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
rminus_wf,
member_rioint_lemma,
member_roiint_lemma,
rless-implies-rless,
int-to-real_wf,
i-member_wf,
roiint_wf,
arctangent-rinv,
req_witness,
arctangent_wf,
rinv_wf2,
sq_stable__rless,
rless_wf,
rsub_wf,
halfpi_wf,
set_wf,
real_wf,
rioint_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req_functionality,
arctangent_functionality,
req_inversion,
rinv-rminus,
req_weakening,
arctangent-rminus,
rsub_functionality,
rmul_preserves_req,
rless-int,
req-implies-req,
rmul_wf,
itermMultiply_wf,
real_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
dependent_set_memberEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
inlFormation,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
productElimination,
approximateComputation,
int_eqEquality,
intEquality,
applyLambdaEquality,
applyEquality,
setEquality,
inrFormation,
minusEquality,
independent_pairFormation
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (-\minfty{}, r0)\} ]. (arctangent(rinv(x)) = (-(\mpi{}/2) - arctangent(x)))
Date html generated:
2018_05_22-PM-03_04_19
Last ObjectModification:
2017_10_22-PM-09_40_11
Theory : reals_2
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