Nuprl Lemma : rcos_wf1
∀[x:ℝ]. (rcos(x) ∈ {y:ℝ| cosine(x) = y} )
Proof
Definitions occuring in Statement :
rcos: rcos(x),
cosine: cosine(x),
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
rcos: rcos(x),
subtype_rel: A ⊆r B,
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
approx-arg_wf,
cosine_wf,
real_wf,
i-member_wf,
riiint_wf,
rminus_wf,
sine_wf,
req_functionality,
rminus_functionality,
sine_functionality,
req_weakening,
req_wf,
derivative-cosine,
false_wf,
le_wf,
subtype_rel_sets,
req_inversion,
rleq_wf,
squash_wf,
true_wf,
rabs-rminus,
int-to-real_wf,
iff_weakening_equal,
rabs-sine-rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
setEquality,
because_Cache,
independent_functionElimination,
lambdaFormation,
independent_isectElimination,
productElimination,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
applyEquality,
axiomEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[x:\mBbbR{}]. (rcos(x) \mmember{} \{y:\mBbbR{}| cosine(x) = y\} )
Date html generated:
2017_01_09-AM-09_10_43
Last ObjectModification:
2016_11_25-PM-09_54_37
Theory : reals_2
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