Nuprl Lemma : sinh-rless
∀x,y:ℝ. sinh(x) < sinh(y) supposing x < y
Proof
Definitions occuring in Statement :
sinh: sinh(x),
rless: x < y,
real: ℝ,
uimplies: b supposing a,
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,
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
top: Top,
true: True,
uiff: uiff(P;Q),
and: P ∧ Q,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
rge: x ≥ y,
guard: {T}
Lemmas referenced :
derivative-implies-strictly-increasing,
riiint_wf,
iproper-riiint,
sinh_wf,
real_wf,
i-member_wf,
cosh_wf,
derivative-sinh,
set_wf,
member_riiint_lemma,
true_wf,
rless-implies-rless,
rless_wf,
rsub_wf,
itermSubtract_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
int-to-real_wf,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
function-is-continuous,
req_functionality,
cosh_functionality,
req_weakening,
req_wf,
rless-int,
rless_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,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_isectElimination,
productElimination,
approximateComputation,
int_eqEquality,
intEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed
Latex:
\mforall{}x,y:\mBbbR{}. sinh(x) < sinh(y) supposing x < y
Date html generated:
2017_10_04-PM-10_46_41
Last ObjectModification:
2017_06_24-PM-00_32_46
Theory : reals_2
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