Nuprl Lemma : arctangent-rtan
∀[x:{x:ℝ| x ∈ (-(π/2), π/2)} ]. (arctangent(rtan(x)) = x)
Proof
Definitions occuring in Statement :
arctangent: arctangent(x),
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 :
cand: A c∧ B,
exists: ∃x:A. B[x],
rtan: rtan(x),
rdiv: (x/y),
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
req_int_terms: t1 ≡ t2,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q,
rneq: x ≠ y,
rge: x ≥ y,
true: True,
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
guard: {T},
so_apply: x[s],
prop: ℙ,
rfun: I ⟶ℝ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
rtan0,
arctangent_functionality,
arctangent0,
real_term_value_minus_lemma,
itermMinus_wf,
halfpi-positive,
rmul-zero-both,
rless_functionality,
rmul_reverses_rless_iff,
member_rooint_lemma,
rsin-rcos-pythag,
radd_comm,
rmul-rinv,
rnexp-rdiv,
req_inversion,
rsin_wf,
real_term_value_add_lemma,
real_term_value_mul_lemma,
rinv-as-rdiv,
rmul-rinv3,
req_transitivity,
rmul-identity1,
itermAdd_wf,
itermConstant_wf,
itermMultiply_wf,
rinv_wf2,
rmul_preserves_req,
derivative_functionality,
rmul_wf,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermVar_wf,
itermSubtract_wf,
member_riiint_lemma,
all_wf,
rleq_wf,
rleq_weakening,
rtan_functionality_wrt_rleq,
rleq_functionality_wrt_implies,
monotone-maps-compact,
rdiv_functionality,
radd_functionality,
rcos_functionality,
rnexp_functionality,
req_functionality,
req_weakening,
derivative-arctangent,
derivative-rtan,
req_wf,
iproper-riiint,
rless_wf,
rdiv_wf,
riiint_wf,
chain-rule,
radd_functionality_wrt_rleq,
rleq_weakening_equal,
rless_functionality_wrt_implies,
rless-int,
trivial-rless-radd,
rnexp_wf,
radd_wf,
le_wf,
false_wf,
rcos_wf,
rnexp-positive,
rcos-positive,
rnexp2-nonneg,
set_wf,
req_witness,
derivative-id,
rtan_wf,
arctangent_wf,
i-member_wf,
real_wf,
int-to-real_wf,
halfpi-interval-proper,
halfpi_wf,
rminus_wf,
rooint_wf,
antiderivatives-equal
Rules used in proof :
productEquality,
minusEquality,
dependent_pairFormation,
intEquality,
int_eqEquality,
approximateComputation,
voidEquality,
voidElimination,
isect_memberEquality,
functionEquality,
inlFormation,
inrFormation,
equalitySymmetry,
equalityTransitivity,
baseClosed,
imageMemberEquality,
independent_isectElimination,
productElimination,
independent_pairFormation,
dependent_set_memberEquality,
lambdaFormation,
isect_memberFormation,
rename,
setElimination,
because_Cache,
hypothesisEquality,
setEquality,
natural_numberEquality,
lambdaEquality,
sqequalRule,
independent_functionElimination,
hypothesis,
isectElimination,
thin,
dependent_functionElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (-(\mpi{}/2), \mpi{}/2)\} ]. (arctangent(rtan(x)) = x)
Date html generated:
2018_05_22-PM-03_02_23
Last ObjectModification:
2018_05_20-PM-11_09_57
Theory : reals_2
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