Nuprl Lemma : rcos-radd
∀[x,y:ℝ]. (rcos(x + y) = ((rcos(x) * rcos(y)) - rsin(x) * rsin(y)))
Proof
Definitions occuring in Statement :
rcos: rcos(x),
rsin: rsin(x),
rsub: x - y,
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,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
prop: ℙ,
so_apply: x[s],
rfun-eq: rfun-eq(I;f;g),
all: ∀x:A. B[x],
top: Top,
true: True,
rcos: rcos(x),
approx-arg: approx-arg(f;B;x),
accelerate: accelerate(k;f),
rsub: x - y,
radd: a + b,
r-ap: f(x),
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A
Lemmas referenced :
derivative_unique,
riiint_wf,
iproper-riiint,
rsin_wf,
radd_wf,
real_wf,
i-member_wf,
rcos_wf,
rsub_wf,
rmul_wf,
member_riiint_lemma,
true_wf,
req_witness,
rcos_functionality,
req_functionality,
req_weakening,
deriviative-rsin,
req_wf,
derivative-function-radd-const,
deriviative-rcos,
derivative-const-mul2,
derivative-add,
rminus_wf,
set_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermMinus_wf,
int-to-real_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
derivative_functionality,
rsin-radd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_functionElimination,
sqequalRule,
lambdaEquality,
setElimination,
rename,
hypothesisEquality,
setEquality,
because_Cache,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
productElimination,
independent_isectElimination,
lambdaFormation,
computeAll,
int_eqEquality,
intEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. (rcos(x + y) = ((rcos(x) * rcos(y)) - rsin(x) * rsin(y)))
Date html generated:
2017_10_04-PM-10_21_35
Last ObjectModification:
2017_07_28-AM-08_48_19
Theory : reals_2
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