Nuprl Lemma : permutation-s-group_wf
∀[rv:SeparationSpace]. ∀[sepw:∀x:Point. ∀y:{y:Point| x # y} . x # y]. (Perm(rv) ∈ s-Group)
Proof
Definitions occuring in Statement :
permutation-s-group: Perm(rv),
s-group: s-Group,
ss-sep: x # y,
ss-point: Point,
separation-space: SeparationSpace,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
exists: ∃x:A. B[x],
or: P ∨ Q,
fun-sep: fun-sep(ss;A;f;g),
pi1: fst(t),
pi2: snd(t),
uimplies: b supposing a,
uiff: uiff(P;Q),
permutation-s-group: Perm(rv),
false: False,
not: ¬A,
ss-eq: x ≡ y,
guard: {T},
compose: f o g,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
top: Top,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
permutation-ss-sep,
exists_wf,
or_wf,
permutation-ss-eq-iff,
mk-s-group_wf,
ss-eq_transitivity,
compose_wf,
separation-space_wf,
permutation-ss_wf,
ss-eq_wf,
all_wf,
ss-sep_wf,
ss-eq_weakening,
ss-point_wf,
permutation-ss-point
Rules used in proof :
inrEquality,
dependent_pairEquality,
inlEquality,
unionElimination,
independent_isectElimination,
setEquality,
axiomEquality,
equalitySymmetry,
equalityTransitivity,
rename,
setElimination,
functionEquality,
functionExtensionality,
applyEquality,
productEquality,
productElimination,
independent_pairFormation,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
lambdaFormation,
hypothesisEquality,
lambdaEquality,
independent_pairEquality,
dependent_set_memberEquality,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
sqequalRule,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:SeparationSpace]. \mforall{}[sepw:\mforall{}x:Point. \mforall{}y:\{y:Point| x \# y\} . x \# y]. (Perm(rv) \mmember{} s-Group)
Date html generated:
2016_11_08-AM-09_13_00
Last ObjectModification:
2016_11_03-AM-11_22_25
Theory : inner!product!spaces
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