Nuprl Lemma : rabs-cosine-rleq
∀x:ℝ. (|cosine(x)| ≤ r1)
Proof
Definitions occuring in Statement :
cosine: cosine(x),
rleq: x ≤ y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
nat: ℕ,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
exp: i^n,
primrec: primrec(n;b;c),
subtract: n - m,
subtype_rel: A ⊆r B,
guard: {T},
sq_type: SQType(T),
rsub: x - y
Lemmas referenced :
sine-cosine-pythag,
real_wf,
rnexp-rleq-iff,
rabs_wf,
cosine_wf,
int-to-real_wf,
zero-rleq-rabs,
rleq-int,
false_wf,
less_than_wf,
rnexp_wf,
le_wf,
rnexp2-nonneg,
rleq_functionality,
req_inversion,
rabs-rnexp,
req_weakening,
rabs-of-nonneg,
exp_wf2,
subtype_base_sq,
nat_plus_wf,
set_subtype_base,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
exp-one,
iff_weakening_equal,
radd-preserves-req,
radd_wf,
sine_wf,
rminus_wf,
req_wf,
rmul_wf,
rnexp-int,
uiff_transitivity,
req_functionality,
req_transitivity,
radd_functionality,
rminus-as-rmul,
radd-assoc,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
radd_comm,
radd-zero-both,
radd-preserves-rleq,
rsub_wf,
rleq_wf,
radd-ac,
radd-rminus-both,
radd-rminus-assoc
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
natural_numberEquality,
independent_functionElimination,
productElimination,
sqequalRule,
independent_pairFormation,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
because_Cache,
independent_isectElimination,
instantiate,
cumulativity,
intEquality,
lambdaEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
minusEquality,
addEquality
Latex:
\mforall{}x:\mBbbR{}. (|cosine(x)| \mleq{} r1)
Date html generated:
2017_10_04-PM-10_20_53
Last ObjectModification:
2017_07_28-AM-08_48_11
Theory : reals_2
Home
Index