Nuprl Lemma : derivative-sinh
d(sinh(x))/dx = λx.cosh(x) on (-∞, ∞)
Proof
Definitions occuring in Statement :
sinh: sinh(x),
cosh: cosh(x),
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞)
Definitions unfolded in proof :
cosh: cosh(x),
sinh: sinh(x),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
prop: ℙ,
rfun: I ⟶ℝ,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
sq_type: SQType(T),
false: False,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
derivative-rexp-fun2,
rminus_wf,
real_wf,
int-to-real_wf,
req_weakening,
req_wf,
derivative-rminus,
riiint_wf,
rexp_wf,
i-member_wf,
rmul_wf,
set_wf,
rmul_comm,
derivative_functionality,
rdiv_wf,
rless-int,
rless_wf,
rsub_wf,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
expr_wf,
radd_wf,
rmul_preserves_req,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
req-iff-rsub-is-0,
minus-one-mul,
itermMinus_wf,
derivative-const-mul,
derivative-sub,
derivative-rexp,
req_functionality,
req_transitivity,
int-rdiv-req,
rdiv_functionality,
rsub_functionality,
expr-req,
radd_functionality,
rmul-rinv3,
int-rinv-cancel,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
isectElimination,
hypothesisEquality,
hypothesis,
minusEquality,
natural_numberEquality,
independent_functionElimination,
lambdaFormation,
because_Cache,
independent_isectElimination,
setElimination,
rename,
setEquality,
inrFormation,
productElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
addLevel,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
applyEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidEquality
Latex:
d(sinh(x))/dx = \mlambda{}x.cosh(x) on (-\minfty{}, \minfty{})
Date html generated:
2017_10_04-PM-10_46_31
Last ObjectModification:
2017_06_24-PM-00_20_07
Theory : reals_2
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