Nuprl Lemma : arctan_wf1
∀[x:ℝ]. (arctan(x) ∈ {y:ℝ| y = arctangent(x)} )
Proof
Definitions occuring in Statement :
arctan: arctan(x),
arctangent: arctangent(x),
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
true: True,
rge: x ≥ y,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
sq_stable: SqStable(P),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top,
arctan: arctan(x)
Lemmas referenced :
rnexp2-nonneg,
real_wf,
approx-arg_wf,
full-arctan_wf,
req_wf,
arctangent_wf,
i-member_wf,
riiint_wf,
rdiv_wf,
int-to-real_wf,
radd_wf,
rnexp_wf,
false_wf,
le_wf,
rless_wf,
req_functionality,
rdiv_functionality,
req_weakening,
radd_functionality,
rnexp_functionality,
trivial-rless-radd,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
derivative-arctangent,
set_wf,
derivative_functionality,
equal_wf,
sq_stable__req,
req_inversion,
rabs_wf,
rmul_preserves_rleq,
rmul_wf,
rmul-zero-both,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
req-iff-rsub-is-0,
rleq-int,
rmul-identity1,
trivial-rleq-radd,
rleq_functionality,
rabs-of-nonneg,
req_transitivity,
rmul-rinv,
real_polynomial_null,
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
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isect_memberFormation,
sqequalRule,
lambdaEquality,
isectElimination,
setElimination,
rename,
applyEquality,
setEquality,
natural_numberEquality,
because_Cache,
dependent_set_memberEquality,
independent_pairFormation,
independent_isectElimination,
inrFormation,
independent_functionElimination,
productElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[x:\mBbbR{}]. (arctan(x) \mmember{} \{y:\mBbbR{}| y = arctangent(x)\} )
Date html generated:
2018_05_22-PM-03_07_13
Last ObjectModification:
2017_10_27-AM-01_02_31
Theory : reals_2
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