Nuprl Lemma : sinh-inv-sinh
∀[x:ℝ]. (sinh(inv-sinh(x)) = x)
Proof
Definitions occuring in Statement :
inv-sinh: inv-sinh(x),
sinh: sinh(x),
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
inv-sinh: inv-sinh(x),
sinh: sinh(x),
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top,
sq_stable: SqStable(P)
Lemmas referenced :
inv-sinh-domain,
req_witness,
sinh_wf,
inv-sinh_wf,
real_wf,
ln_wf,
radd_wf,
rsqrt_wf,
rmul_wf,
int-to-real_wf,
rleq_wf,
req_wf,
rless_wf,
set_wf,
rlog_wf,
expr_wf,
rexp_wf,
rminus_wf,
equal_wf,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
rsub_wf,
rdiv_wf,
rless-int,
rmul_preserves_req,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
req-iff-rsub-is-0,
minus-one-mul,
itermMinus_wf,
rmul_comm,
req_functionality,
int-rdiv-req,
req_weakening,
req_transitivity,
radd_functionality,
int-rinv-cancel,
rmul-rinv3,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
sq_stable__req,
rexp-positive,
uiff_transitivity,
rexp_functionality,
rexp-rlog,
rexp-rminus,
rdiv_functionality,
req_inversion,
rless_transitivity1,
rleq_weakening,
rminus_functionality,
rmul-rinv,
rmul_functionality,
radd-preserves-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
independent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
productElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
sqequalRule,
because_Cache,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
addLevel,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
voidElimination,
baseClosed,
inrFormation,
independent_pairFormation,
imageMemberEquality,
minusEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidEquality,
imageElimination
Latex:
\mforall{}[x:\mBbbR{}]. (sinh(inv-sinh(x)) = x)
Date html generated:
2017_10_04-PM-10_44_44
Last ObjectModification:
2017_06_24-AM-10_48_57
Theory : reals_2
Home
Index