Nuprl Lemma : permutation-ss-eq-iff
∀[rv:SeparationSpace]. ∀[f,g,f',g':Point ⟶ Point]. uiff(<f, g> ≡ <f', g'>;(∀x:Point. f x ≡ f' x) ∧ (∀x:Point. g x ≡ g'\000C x))
Proof
Definitions occuring in Statement :
permutation-ss: permutation-ss(ss),
ss-eq: x ≡ y,
ss-point: Point,
separation-space: SeparationSpace,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>
Definitions unfolded in proof :
guard: {T},
false: False,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
exists: ∃x:A. B[x],
or: P ∨ Q,
fun-sep: fun-sep(ss;A;f;g),
implies: P ⇒ Q,
not: ¬A,
ss-eq: x ≡ y,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
top: Top,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
separation-space_wf,
ss-eq_wf,
all_wf,
fun-sep_wf,
or_wf,
not_wf,
exists_wf,
ss-point_wf,
ss-sep_wf,
permutation-ss-eq
Rules used in proof :
functionEquality,
productEquality,
unionElimination,
dependent_functionElimination,
independent_pairEquality,
productElimination,
because_Cache,
inrFormation,
lambdaEquality,
functionExtensionality,
applyEquality,
hypothesisEquality,
dependent_pairFormation,
inlFormation,
independent_functionElimination,
lambdaFormation,
independent_pairFormation,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalRule,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:SeparationSpace]. \mforall{}[f,g,f',g':Point {}\mrightarrow{} Point].
uiff(<f, g> \mequiv{} <f', g'>(\mforall{}x:Point. f x \mequiv{} f' x) \mwedge{} (\mforall{}x:Point. g x \mequiv{} g' x))
Date html generated:
2016_11_08-AM-09_12_50
Last ObjectModification:
2016_11_03-AM-00_27_44
Theory : inner!product!spaces
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