Nuprl Lemma : full-arctan_wf
∀[x:ℝ]. (full-arctan(x) ∈ {y:ℝ| y = arctangent(x)} )
Proof
Definitions occuring in Statement :
full-arctan: full-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 :
rminus: -(x),
full-arctan: full-arctan(x),
top: Top,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
rgt: x > y,
rge: x ≥ y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rneq: x ≠ y,
or: P ∨ Q,
sq_exists: ∃x:A [B[x]],
rless: x < y,
real: ℝ,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
prop: ℙ,
false: False,
guard: {T},
all: ∀x:A. B[x],
sq_type: SQType(T),
uimplies: b supposing a,
implies: P ⇒ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
member: t ∈ T,
true: True,
int-to-real: r(n),
int-rdiv: (a)/k1,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
cand: A c∧ B,
and: P ∧ Q
Lemmas referenced :
arctangent-rminus,
rminus-rminus,
rminus_functionality_wrt_rleq,
rleq_functionality_wrt_implies,
arctangent-reduction-1,
rless-implies-rless,
rless_transitivity2,
rmul_preserves_rless,
radd-preserves-rleq,
real_term_value_minus_lemma,
int-rinv-cancel2,
int-rinv-cancel,
radd_functionality,
minus-one-mul-top,
itermMinus_wf,
rdiv_functionality,
rminus_functionality,
rleq_wf,
rminus_wf,
rabs-rleq-iff,
real_term_value_add_lemma,
rabs_functionality,
itermAdd_wf,
rabs-rdiv,
radd_functionality_wrt_rless1,
trivial-rless-radd,
radd_wf,
arctangent-rinv,
rinv-as-rdiv,
arctangent_functionality,
req_inversion,
2-MachinPi4,
rsub_functionality,
req_functionality,
uiff_transitivity,
member_roiint_lemma,
halfpi_wf,
pi_wf,
MachinPi4_wf,
int-rmul_wf,
rsub_wf,
rmul-rinv3,
rmul-int,
rinv-mul-as-rdiv,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-rinv,
req_transitivity,
rabs-of-nonneg,
rleq_functionality,
rmul_comm,
false_wf,
rleq-int,
req-iff-rsub-is-0,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
rmul-zero-both,
rmul_wf,
rmul_preserves_rleq,
rabs_wf,
arctangent_wf,
req_wf,
set_wf,
atan-small_wf,
rleq_weakening_rless,
rleq_weakening_equal,
rless_functionality_wrt_implies,
req_weakening,
int-rdiv-req,
rless_functionality,
rless-int-fractions2,
rless-int,
rdiv_wf,
equal_wf,
rless_wf,
or_wf,
rless-case_wf,
real_wf,
less_than_wf,
int-to-real_wf,
nequal_wf,
true_wf,
equal-wf-base,
int_subtype_base,
subtype_base_sq,
int-rdiv_wf
Rules used in proof :
axiomEquality,
isect_memberFormation,
productEquality,
setEquality,
voidEquality,
isect_memberEquality,
int_eqEquality,
approximateComputation,
multiplyEquality,
inrFormation,
unionElimination,
productElimination,
rename,
setElimination,
lambdaEquality,
imageMemberEquality,
because_Cache,
minusEquality,
hypothesisEquality,
baseClosed,
voidElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
hypothesis,
independent_isectElimination,
intEquality,
cumulativity,
instantiate,
lambdaFormation,
addLevel,
dependent_set_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
applyEquality,
addEquality,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut
Latex:
\mforall{}[x:\mBbbR{}]. (full-arctan(x) \mmember{} \{y:\mBbbR{}| y = arctangent(x)\} )
Date html generated:
2018_05_22-PM-03_07_02
Last ObjectModification:
2018_05_20-PM-11_27_36
Theory : reals_2
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