Nuprl Lemma : sg-subgroup_wf
∀[sg:s-Group]. ∀[P:Point ⟶ ℙ]. (sg-subgroup(sg;x.P[x]) ∈ ℙ)
Proof
Definitions occuring in Statement :
sg-subgroup: sg-subgroup(sg;x.P[x]),
s-group: s-Group,
ss-point: Point,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
sg-subgroup: sg-subgroup(sg;x.P[x]),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
s-group_wf,
sg-op_wf,
sg-inv_wf,
all_wf,
sg-id_wf,
s-group_subtype1,
ss-point_wf
Rules used in proof :
isect_memberEquality,
universeEquality,
cumulativity,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
functionEquality,
lambdaEquality,
because_Cache,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesisEquality,
functionExtensionality,
applyEquality,
productEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[sg:s-Group]. \mforall{}[P:Point {}\mrightarrow{} \mBbbP{}]. (sg-subgroup(sg;x.P[x]) \mmember{} \mBbbP{})
Date html generated:
2016_11_08-AM-09_12_30
Last ObjectModification:
2016_11_03-PM-00_22_14
Theory : inner!product!spaces
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