Nuprl Lemma : rtan-rminus
∀[x:{x:ℝ| x ∈ (-(π/2), π/2)} ]. (rtan(-(x)) = -(rtan(x)))
Proof
Definitions occuring in Statement :
rtan: rtan(x),
halfpi: π/2,
rooint: (l, u),
i-member: r ∈ I,
req: x = y,
rminus: -(x),
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
rtan: rtan(x),
rneq: x ≠ y,
or: P ∨ Q,
rat_term_to_real: rat_term_to_real(f;t),
rtermMinus: rtermMinus(num),
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
pi2: snd(t),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
member_rooint_lemma,
istype-void,
req_witness,
rtan_wf,
rminus_wf,
rless-implies-rless,
halfpi_wf,
rless_wf,
real_wf,
i-member_wf,
rooint_wf,
rsub_wf,
itermSubtract_wf,
itermVar_wf,
itermMinus_wf,
req-iff-rsub-is-0,
real_polynomial_null,
int-to-real_wf,
istype-int,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma,
rcos-positive,
rdiv_wf,
rsin_wf,
rcos_wf,
assert-rat-term-eq2,
rtermDivide_wf,
rtermMinus_wf,
rtermVar_wf,
req_functionality,
rdiv_functionality,
rsin-rminus,
rcos-rminus,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
setElimination,
thin,
rename,
sqequalHypSubstitution,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
hypothesis,
isectElimination,
sqequalRule,
dependent_set_memberEquality_alt,
hypothesisEquality,
productElimination,
independent_isectElimination,
independent_pairFormation,
because_Cache,
productIsType,
universeIsType,
independent_functionElimination,
setIsType,
natural_numberEquality,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
inrFormation_alt
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (-(\mpi{}/2), \mpi{}/2)\} ]. (rtan(-(x)) = -(rtan(x)))
Date html generated:
2019_10_30-AM-11_44_22
Last ObjectModification:
2019_04_03-AM-00_21_32
Theory : reals_2
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