Nuprl Lemma : arccos_wf
∀[a:{a:ℝ| a ∈ [r(-1), r1]} ]. (arccos(a) ∈ {x:ℝ| (x ∈ [r0, π]) ∧ (rcos(x) = a)} )
Proof
Definitions occuring in Statement :
arccos: arccos(x),
pi: π,
rcos: rcos(x),
rccint: [l, u],
i-member: r ∈ I,
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
all: ∀x:A. B[x],
top: Top,
arccos: arccos(x),
and: P ∧ Q,
pi: π,
prop: ℙ,
cand: A c∧ B,
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B
Lemmas referenced :
arcsin_wf,
member_rccint_lemma,
istype-void,
rsub_wf,
halfpi_wf,
i-member_wf,
rccint_wf,
int-to-real_wf,
rleq-implies-rleq,
rleq_wf,
int-rmul_wf,
req_wf,
rcos_wf,
real_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
rmul_wf,
rminus_wf,
itermMinus_wf,
itermMultiply_wf,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_functionality,
req_weakening,
int-rmul-req,
real_term_value_minus_lemma,
real_term_value_mul_lemma,
radd_wf,
itermAdd_wf,
rsin_wf,
rminus-rminus,
req_functionality,
radd_functionality,
arcsin-rminus,
real_term_value_add_lemma,
uiff_transitivity,
rcos_functionality,
rcos-shift-half-pi,
rminus_functionality,
rsin-arcsin
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyLambdaEquality,
setElimination,
rename,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
productElimination,
dependent_set_memberEquality_alt,
universeIsType,
minusEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_isectElimination,
independent_pairFormation,
productIsType,
setIsType,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
applyEquality,
inhabitedIsType,
independent_functionElimination
Latex:
\mforall{}[a:\{a:\mBbbR{}| a \mmember{} [r(-1), r1]\} ]. (arccos(a) \mmember{} \{x:\mBbbR{}| (x \mmember{} [r0, \mpi{}]) \mwedge{} (rcos(x) = a)\} )
Date html generated:
2019_10_31-AM-06_16_03
Last ObjectModification:
2019_05_23-AM-11_31_21
Theory : reals_2
Home
Index