Nuprl Lemma : arccos_functionality
∀[x:{x:ℝ| x ∈ [r(-1), r1]} ]. ∀[x':ℝ]. arccos(x) = arccos(x') supposing x = x'
Proof
Definitions occuring in Statement :
arccos: arccos(x),
rccint: [l, u],
i-member: r ∈ I,
req: x = y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
prop: ℙ,
arccos: arccos(x),
subtype_rel: A ⊆r B,
i-member: r ∈ I,
rccint: [l, u],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
member_rccint_lemma,
istype-void,
rleq_transitivity,
int-to-real_wf,
rleq_weakening,
req_inversion,
rleq_wf,
sq_stable__req,
rsub_wf,
halfpi_wf,
arcsin_wf,
subtype_rel_self,
real_wf,
i-member_wf,
rccint_wf,
req_wf,
req_weakening,
req_functionality,
rsub_functionality,
arcsin_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
setElimination,
thin,
rename,
sqequalHypSubstitution,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
hypothesis,
dependent_set_memberEquality_alt,
hypothesisEquality,
productElimination,
isectElimination,
minusEquality,
natural_numberEquality,
independent_isectElimination,
because_Cache,
independent_pairFormation,
sqequalRule,
productIsType,
universeIsType,
applyEquality,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
setEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
setIsType
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} [r(-1), r1]\} ]. \mforall{}[x':\mBbbR{}]. arccos(x) = arccos(x') supposing x = x'
Date html generated:
2019_10_31-AM-06_16_35
Last ObjectModification:
2019_05_24-PM-02_12_56
Theory : reals_2
Home
Index