Nuprl Lemma : arctangent-rminus
∀[x:ℝ]. (arctangent(-(x)) = -(arctangent(x)))
Proof
Definitions occuring in Statement :
arctangent: arctangent(x),
req: x = y,
rminus: -(x),
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
cand: A c∧ B,
uimplies: b supposing a,
implies: P ⇒ Q,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
arctangent-bounds,
rminus_wf,
member_rooint_lemma,
rtan_one_one,
arctangent_wf,
rless_wf,
halfpi_wf,
rless-implies-rless,
req_witness,
real_wf,
rsub_wf,
itermSubtract_wf,
itermVar_wf,
itermMinus_wf,
req-iff-rsub-is-0,
real_polynomial_null,
int-to-real_wf,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma,
rtan_wf,
req_weakening,
req_functionality,
rtan-arctangent,
req_transitivity,
rtan-rminus,
rminus_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
sqequalRule,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
natural_numberEquality,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality
Latex:
\mforall{}[x:\mBbbR{}]. (arctangent(-(x)) = -(arctangent(x)))
Date html generated:
2018_05_22-PM-03_04_00
Last ObjectModification:
2017_10_22-PM-08_25_32
Theory : reals_2
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