Nuprl Lemma : sinh-rleq
∀[x,y:ℝ]. sinh(x) ≤ sinh(y) supposing x ≤ y
Proof
Definitions occuring in Statement :
sinh: sinh(x),
rleq: x ≤ y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[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,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
guard: {T},
top: Top,
true: True,
increasing-on-interval: f[x] increasing for x ∈ I,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
rge: x ≥ y
Lemmas referenced :
derivative-implies-increasing,
riiint_wf,
iproper-riiint,
sinh_wf,
real_wf,
i-member_wf,
cosh_wf,
derivative-sinh,
set_wf,
less_than'_wf,
rsub_wf,
nat_plus_wf,
rleq_wf,
member_riiint_lemma,
true_wf,
function-is-continuous,
req_functionality,
cosh_functionality,
req_weakening,
req_wf,
int-to-real_wf,
rleq-int,
false_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
cosh-ge-1
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,
isect_memberFormation,
productElimination,
independent_pairEquality,
applyEquality,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
independent_isectElimination,
independent_pairFormation
Latex:
\mforall{}[x,y:\mBbbR{}]. sinh(x) \mleq{} sinh(y) supposing x \mleq{} y
Date html generated:
2017_10_04-PM-10_46_35
Last ObjectModification:
2017_06_24-PM-00_30_04
Theory : reals_2
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