Nuprl Lemma : rtan-rsub
∀[x,y:{x:ℝ| x ∈ (-(π/2), π/2)} ].
rtan(x - y) = (rtan(x) - rtan(y)/r1 + (rtan(x) * rtan(y))) supposing x - y ∈ (-(π/2), π/2)
Proof
Definitions occuring in Statement :
rtan: rtan(x),
halfpi: π/2,
rooint: (l, u),
i-member: r ∈ I,
rdiv: (x/y),
rsub: x - y,
req: x = y,
rmul: a * b,
rminus: -(x),
radd: a + b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
top: Top,
subtype_rel: A ⊆r B,
guard: {T},
i-member: r ∈ I,
rooint: (l, u),
rsub: x - y,
rtan: rtan(x),
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermSubtract: left "-" right,
rtermAdd: left "+" right,
rtermConstant: "const",
rtermMultiply: left "*" right,
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
rtermMinus: rtermMinus(num),
pi2: snd(t),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rcos-positive,
radd_wf,
i-member_wf,
rooint_wf,
rminus_wf,
halfpi_wf,
real_wf,
sq_stable__req,
rtan_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rmul_wf,
member_rooint_lemma,
istype-void,
rless-implies-rless,
rless_wf,
subtype_rel_self,
rcos_wf,
rsin_wf,
rmul_preserves_rless,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
rinv_wf2,
itermAdd_wf,
itermMinus_wf,
req-iff-rsub-is-0,
rless_functionality,
req_weakening,
rcos-radd,
req_transitivity,
radd_functionality,
rminus_functionality,
rmul-rinv3,
rinv-mul-as-rdiv,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
rsub_functionality,
rmul_functionality,
rtan-rminus,
rtan-radd,
assert-rat-term-eq2,
rtermDivide_wf,
rtermAdd_wf,
rtermVar_wf,
rtermMinus_wf,
rtermSubtract_wf,
rtermConstant_wf,
rtermMultiply_wf,
req_functionality,
rneq_functionality,
rdiv_functionality
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
independent_pairFormation,
setElimination,
rename,
dependent_set_memberEquality_alt,
isectElimination,
universeIsType,
inhabitedIsType,
setIsType,
closedConclusion,
natural_numberEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
productElimination,
productIsType,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
inrFormation_alt,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
promote_hyp
Latex:
\mforall{}[x,y:\{x:\mBbbR{}| x \mmember{} (-(\mpi{}/2), \mpi{}/2)\} ].
rtan(x - y) = (rtan(x) - rtan(y)/r1 + (rtan(x) * rtan(y))) supposing x - y \mmember{} (-(\mpi{}/2), \mpi{}/2)
Date html generated:
2019_10_30-AM-11_44_28
Last ObjectModification:
2019_04_03-AM-00_21_23
Theory : reals_2
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