Nuprl Lemma : arccos-unique
∀[x:{x:ℝ| x ∈ [r(-1), r1]} ]. ∀[y:{y:ℝ| y ∈ [r0, π]} ]. ((rcos(y) = x) ⇒ (arccos(x) = y))
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],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
top: Top,
uimplies: b supposing a,
not: ¬A,
rneq: x ≠ y,
or: P ∨ Q,
strictly-decreasing-on-interval: f[x] strictly-decreasing for x ∈ I,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
cand: A c∧ B,
guard: {T},
false: False
Lemmas referenced :
rcos-arccos,
req_wf,
rcos_wf,
req_witness,
arccos_wf,
member_rccint_lemma,
istype-void,
real_wf,
i-member_wf,
rccint_wf,
int-to-real_wf,
pi_wf,
rcos-strictly-decreasing,
not-rneq,
rneq_wf,
subtype_rel_sets_simple,
req_inversion,
rless_transitivity1,
rleq_weakening,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
universeIsType,
setElimination,
rename,
hypothesis,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
applyEquality,
isect_memberEquality_alt,
voidElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_functionElimination,
functionIsTypeImplies,
setIsType,
natural_numberEquality,
isectIsTypeImplies,
minusEquality,
independent_isectElimination,
unionElimination,
productEquality,
productElimination,
productIsType
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} [r(-1), r1]\} ]. \mforall{}[y:\{y:\mBbbR{}| y \mmember{} [r0, \mpi{}]\} ]. ((rcos(y) = x) {}\mRightarrow{} (arccos(x) = y))
Date html generated:
2019_10_31-AM-06_16_27
Last ObjectModification:
2019_05_23-AM-11_40_38
Theory : reals_2
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