Nuprl Lemma : mk-s-subgroup_wf
∀[sg:s-Group]. ∀[P:Point ⟶ ℙ]. mk-s-subgroup(sg;x.P[x]) ∈ s-Group supposing sg-subgroup(sg;x.P[x])
Proof
Definitions occuring in Statement :
mk-s-subgroup: mk-s-subgroup(sg;x.P[x]),
sg-subgroup: sg-subgroup(sg;x.P[x]),
s-group: s-Group,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sg-subgroup: sg-subgroup(sg;x.P[x]),
and: P ∧ Q,
mk-s-subgroup: mk-s-subgroup(sg;x.P[x]),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
s-group: s-Group,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
cand: A c∧ B,
s-group-structure: s-GroupStructure,
record+: record+,
record-select: r.x,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
sg-inv: x^-1,
sg-op: (x y),
or: P ∨ Q
Lemmas referenced :
mk-s-group_wf,
set-ss_wf,
ss-point_wf,
set-ss-point,
sg-id_wf,
sg-inv_wf,
sg-subgroup_wf,
s-group-structure_subtype1,
s-group_subtype1,
subtype_rel_transitivity,
s-group_wf,
s-group-structure_wf,
separation-space_wf,
sg-op_wf,
sg-assoc,
set_wf,
sg-op-id,
sg-op-inv,
all_wf,
ss-eq_wf,
subtype_rel_self,
subtype_rel_dep_function,
ss-sep_wf,
or_wf,
set-ss-eq,
set-ss-sep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
functionExtensionality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
setElimination,
rename,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
setEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
independent_isectElimination,
functionEquality,
cumulativity,
lambdaFormation,
independent_pairFormation,
productEquality,
dependentIntersectionElimination,
dependentIntersectionEqElimination
Latex:
\mforall{}[sg:s-Group]. \mforall{}[P:Point {}\mrightarrow{} \mBbbP{}]. mk-s-subgroup(sg;x.P[x]) \mmember{} s-Group supposing sg-subgroup(sg;x.P[x])
Date html generated:
2017_10_02-PM-03_25_13
Last ObjectModification:
2017_07_28-AM-06_57_12
Theory : constructive!algebra
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