Nuprl Lemma : sinh-radd
∀[x,y:ℝ]. (sinh(x + y) = ((sinh(x) * cosh(y)) + (cosh(x) * sinh(y))))
Proof
Definitions occuring in Statement :
sinh: sinh(x),
cosh: cosh(x),
req: x = y,
rmul: a * b,
radd: a + b,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sinh: sinh(x),
cosh: cosh(x),
implies: P ⇒ Q,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
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),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
req_witness,
sinh_wf,
radd_wf,
rmul_wf,
cosh_wf,
real_wf,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
rsub_wf,
expr_wf,
req_wf,
rexp_wf,
rminus_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
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,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
req_weakening,
req_functionality,
int-rdiv-req,
radd_functionality,
rmul_functionality,
req_transitivity,
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,
rminus_functionality,
expr-req,
radd_comm,
uiff_transitivity,
rexp_functionality,
rexp-radd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
addLevel,
lambdaFormation,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
baseClosed,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
inrFormation,
productElimination,
independent_pairFormation,
imageMemberEquality,
minusEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
voidEquality,
int_eqEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. (sinh(x + y) = ((sinh(x) * cosh(y)) + (cosh(x) * sinh(y))))
Date html generated:
2017_10_04-PM-10_41_13
Last ObjectModification:
2017_06_06-PM-00_07_42
Theory : reals_2
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