Nuprl Lemma : cosh2-sinh2
∀[x:ℝ]. ((cosh(x)^2 - sinh(x)^2) = r1)
Proof
Definitions occuring in Statement :
sinh: sinh(x),
cosh: cosh(x),
rnexp: x^k1,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
sinh: sinh(x),
cosh: cosh(x),
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
subtype_rel: A ⊆r B,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
rsub: x - y,
sq_stable: SqStable(P)
Lemmas referenced :
req_witness,
rsub_wf,
rnexp_wf,
false_wf,
le_wf,
cosh_wf,
sinh_wf,
int-to-real_wf,
real_wf,
rmul_wf,
req_functionality,
rsub_functionality,
rnexp2,
req_weakening,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
radd_wf,
expr_wf,
req_wf,
rexp_wf,
rminus_wf,
rdiv_wf,
rless-int,
rless_wf,
rmul_functionality,
int-rdiv-req,
set_wf,
equal_wf,
rmul_preserves_req,
uiff_transitivity,
req_transitivity,
rmul-distrib,
radd_functionality,
rmul_over_rminus,
rminus_functionality,
req_inversion,
rmul-assoc,
rmul-one-both,
radd_comm,
rmul-rdiv-cancel2,
rmul_comm,
rminus-radd,
radd-assoc,
radd-ac,
rdiv_functionality,
rminus-as-rmul,
rminus-rminus,
radd-rminus-both,
radd-zero-both,
rmul-distrib2,
rmul-identity1,
radd-int,
rmul-zero-both,
squash_wf,
iff_weakening_equal,
sq_stable__req,
rexp0,
rexp-radd,
rexp_functionality,
rmul-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
hypothesis,
hypothesisEquality,
because_Cache,
independent_functionElimination,
independent_isectElimination,
productElimination,
addLevel,
instantiate,
cumulativity,
intEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
baseClosed,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
inrFormation,
imageMemberEquality,
minusEquality,
addEquality,
imageElimination,
universeEquality,
multiplyEquality
Latex:
\mforall{}[x:\mBbbR{}]. ((cosh(x)\^{}2 - sinh(x)\^{}2) = r1)
Date html generated:
2017_10_04-PM-10_40_40
Last ObjectModification:
2017_07_28-AM-08_51_04
Theory : reals_2
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