Nuprl Lemma : generated-s-subgroup_wf
∀[sg:s-Group]. ∀[P:Point ⟶ ℙ]. generated-s-subgroup(sg;f.P[f]) ∈ s-Group supposing ∀f:Point. (P[f] ⇒ P[f^-1])
Proof
Definitions occuring in Statement :
generated-s-subgroup: generated-s-subgroup(sg;f.P[f]),
s-group: s-Group,
sg-inv: x^-1,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
generated-s-subgroup: generated-s-subgroup(sg;f.P[f]),
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
s-group: s-Group,
sg-subgroup: sg-subgroup(sg;x.P[x]),
cand: A c∧ B,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
guard: {T},
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q)
Lemmas referenced :
mk-s-subgroup_wf,
exists_wf,
list_wf,
l_all_wf2,
l_member_wf,
ss-point_wf,
ss-eq_wf,
reduce_wf,
sg-op_wf,
sg-id_wf,
all_wf,
s-group-structure_subtype1,
s-group_subtype1,
subtype_rel_transitivity,
s-group_wf,
s-group-structure_wf,
separation-space_wf,
sg-inv_wf,
ss-eq_weakening,
l_all_nil,
reduce_nil_lemma,
nil_wf,
reverse_wf,
map_wf,
l_all_reverse,
l_all_iff,
iff_weakening_equal,
member-map,
reverse-cons,
map_cons_lemma,
reduce_cons_lemma,
reverse_nil_lemma,
map_nil_lemma,
list_induction,
sg-inv-op,
sg-op-id,
sg-op_functionality,
ss-eq_functionality,
uiff_transitivity,
sg-inv-unique,
ss-eq_inversion,
equal_wf,
reduce-append,
sg-inv-inv,
sg-inv-of-op,
ss-eq_transitivity,
sg-assoc,
sg-id-op,
sg-inv-id,
sg-inv_functionality,
l_all_append,
append_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
because_Cache,
hypothesis,
productEquality,
lambdaFormation,
applyEquality,
setElimination,
rename,
functionExtensionality,
setEquality,
independent_isectElimination,
independent_pairFormation,
productElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
functionEquality,
universeEquality,
isect_memberEquality,
cumulativity,
independent_functionElimination,
voidEquality,
voidElimination,
dependent_functionElimination,
dependent_pairFormation,
impliesFunctionality,
addLevel
Latex:
\mforall{}[sg:s-Group]. \mforall{}[P:Point {}\mrightarrow{} \mBbbP{}].
generated-s-subgroup(sg;f.P[f]) \mmember{} s-Group supposing \mforall{}f:Point. (P[f] {}\mRightarrow{} P[f\^{}-1])
Date html generated:
2017_10_02-PM-03_25_27
Last ObjectModification:
2017_07_28-AM-06_57_14
Theory : constructive!algebra
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