Nuprl Lemma : rsin-rsub
∀[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),
rsub: x - y,
req: x = y,
rmul: a * b,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rsub: x - y,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
req_witness,
rsin_wf,
rsub_wf,
rmul_wf,
rcos_wf,
real_wf,
radd_wf,
rminus_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermMinus_wf,
req-iff-rsub-is-0,
req_functionality,
req_transitivity,
rsin-radd,
radd_functionality,
rmul_functionality,
req_weakening,
rsin-rminus,
rcos-rminus,
real_polynomial_null,
int-to-real_wf,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
inhabitedIsType,
sqequalRule,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
universeIsType,
natural_numberEquality,
productElimination,
independent_isectElimination,
dependent_functionElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
voidElimination
Latex:
\mforall{}[x,y:\mBbbR{}]. (rsin(x - y) = ((rsin(x) * rcos(y)) - rcos(x) * rsin(y)))
Date html generated:
2019_10_30-AM-11_41_48
Last ObjectModification:
2019_05_17-AM-11_26_30
Theory : reals_2
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