Nuprl Lemma : rsin-radd
∀[x,y:ℝ]. (rsin(x + y) = ((rsin(x) * rcos(y)) + (rcos(x) * rsin(y))))
Proof
Definitions occuring in Statement :
rcos: rcos(x),
rsin: rsin(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,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
rfun: I ⟶ℝ,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
cand: A c∧ B,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
less_than': less_than'(a;b),
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
infinite-deriv-seq: infinite-deriv-seq(I;i,x.F[i; x]),
le: A ≤ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
eq_int: (i =z j),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
int_upper: {i...},
rge: x ≥ y
Lemmas referenced :
req_witness,
rsin_wf,
radd_wf,
rmul_wf,
rcos_wf,
real_wf,
derivative-function-radd-const,
req_functionality,
rcos_functionality,
req_weakening,
req_wf,
deriviative-rsin,
rminus_wf,
rminus_functionality,
rsin_functionality,
deriviative-rcos,
derivative-minus,
riiint_wf,
i-member_wf,
set_wf,
rminus-rminus,
derivative_functionality,
derivative-add,
derivative-const-mul2,
uiff_transitivity,
radd_functionality,
req_transitivity,
rmul_over_rminus,
rmul_comm,
minus-zero,
rminus-as-rmul,
int-to-real_wf,
squash_wf,
true_wf,
rminus-int,
iff_weakening_equal,
rminus-rminus-eq,
minus-minus,
rmul-zero-both,
radd-zero-both,
rmul_functionality,
rcos0,
rsin0,
radd-zero,
rmul-one-both,
rem_add1,
subtract_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
less_than_wf,
eq_int_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
add-associates,
nat_plus_wf,
add-swap,
add-commutes,
zero-add,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
add-subtract-cancel,
rem_bounds_1,
lelt_wf,
int_seg_wf,
decidable__equal_int,
int_seg_properties,
int_seg_subtype,
int_seg_cases,
nat_wf,
equal-functions-by-Taylor,
rleq_wf,
rabs_wf,
int_upper_wf,
all_wf,
exists_wf,
subtype_rel_self,
rabs-rsin-rleq,
rabs-rcos-rleq,
rabs-rminus,
zero-rleq-rabs,
rleq-int,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq_transitivity,
r-triangle-inequality,
radd_functionality_wrt_rleq,
rleq_weakening,
rabs-rmul,
rmul_functionality_wrt_rleq2,
rleq_functionality,
rmul-int,
radd-int
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_functionElimination,
lambdaEquality,
lambdaFormation,
independent_isectElimination,
productElimination,
setElimination,
rename,
setEquality,
independent_pairFormation,
natural_numberEquality,
minusEquality,
productEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
addLevel,
levelHypothesis,
andLevelFunctionality,
dependent_set_memberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
remainderEquality,
instantiate,
cumulativity,
equalityElimination,
promote_hyp,
addEquality,
hypothesis_subsumption,
inlFormation,
multiplyEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. (rsin(x + y) = ((rsin(x) * rcos(y)) + (rcos(x) * rsin(y))))
Date html generated:
2017_10_04-PM-10_21_31
Last ObjectModification:
2017_07_28-AM-08_48_17
Theory : reals_2
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