Nuprl Lemma : arccos-rminus
∀[a:{a:ℝ| a ∈ [r(-1), r1]} ]. (arccos(-(a)) = (π - arccos(a)))
Proof
Definitions occuring in Statement :
arccos: arccos(x),
pi: π,
rccint: [l, u],
i-member: r ∈ I,
rsub: x - y,
req: x = y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
arccos: arccos(x),
pi: π,
subtype_rel: A ⊆r B,
guard: {T},
i-member: r ∈ I,
rccint: [l, u],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rminus_wf,
sq_stable__rleq,
int-to-real_wf,
member_rccint_lemma,
istype-void,
rleq-implies-rleq,
rleq_wf,
real_wf,
i-member_wf,
rccint_wf,
rsub_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
halfpi_wf,
arcsin_wf,
subtype_rel_self,
int-rmul_wf,
rmul_wf,
itermMultiply_wf,
req_functionality,
rsub_functionality,
req_weakening,
arcsin-rminus,
int-rmul-req,
real_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
dependent_set_memberEquality_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
minusEquality,
natural_numberEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
productElimination,
independent_isectElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
because_Cache,
productIsType,
universeIsType,
setIsType,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setEquality,
inhabitedIsType
Latex:
\mforall{}[a:\{a:\mBbbR{}| a \mmember{} [r(-1), r1]\} ]. (arccos(-(a)) = (\mpi{} - arccos(a)))
Date html generated:
2019_10_31-AM-06_16_43
Last ObjectModification:
2019_05_23-AM-11_45_26
Theory : reals_2
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