Nuprl Lemma : rabs-difference-cosine-rleq
∀x,y:ℝ. (|cosine(x) - cosine(y)| ≤ |x - y|)
Proof
Definitions occuring in Statement :
cosine: cosine(x),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
real: ℝ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
top: Top,
true: True,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
rge: x ≥ y,
rsub: x - y,
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
mean-value-for-bounded-derivative,
riiint_wf,
iproper-riiint,
cosine_wf,
real_wf,
i-member_wf,
rminus_wf,
sine_wf,
req_wf,
set_wf,
derivative-cosine,
int-to-real_wf,
req_weakening,
all_wf,
rleq_wf,
rmul_wf,
rabs_wf,
rsub_wf,
member_riiint_lemma,
true_wf,
radd_wf,
rleq_weakening_equal,
req_functionality,
rminus_functionality,
sine_functionality,
all_functionality_wrt_uimplies,
rleq_functionality_wrt_implies,
rleq_functionality,
rmul-one-both,
squash_wf,
rabs-rminus,
iff_weakening_equal,
rabs-sine-rleq
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
hypothesis,
independent_functionElimination,
sqequalRule,
lambdaEquality,
isectElimination,
setElimination,
rename,
hypothesisEquality,
setEquality,
because_Cache,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}x,y:\mBbbR{}. (|cosine(x) - cosine(y)| \mleq{} |x - y|)
Date html generated:
2016_10_26-PM-00_13_35
Last ObjectModification:
2016_09_12-PM-05_39_47
Theory : reals_2
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