Nuprl Lemma : mk-s-group_wf
∀[ss:SeparationSpace]. ∀[e:Point(ss)]. ∀[i:Point(ss) ⟶ Point(ss)]. ∀[o:{f:Point(ss) ⟶ Point(ss) ⟶ Point(ss)|
(∀x,y,z:Point(ss). f x (f y z) ≡ f (f x y) z)
∧ (∀x:Point(ss). f x e ≡ x)
∧ (∀x:Point(ss). f x (i x) ≡ e)} ].
∀[sep:∀x,x',y,y':Point(ss). (o x y # o x' y' ⇒ (x # x' ∨ y # y'))]. ∀[invsep:∀x,y:Point(ss). (i x # i y ⇒ x # y)].
(mk-s-group(ss; e; i; o; sep; invsep) ∈ s-Group)
Proof
Definitions occuring in Statement :
mk-s-group: mk-s-group(ss; e; i; o; sep; invsep),
s-group: s-Group,
ss-eq: x ≡ y,
ss-sep: x # y,
ss-point: Point(ss),
separation-space: SeparationSpace,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
separation-space: SeparationSpace,
record+: record+,
record-select: r.x,
subtype_rel: A ⊆r B,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
mk-s-group: mk-s-group(ss; e; i; o; sep; invsep),
s-group: s-Group,
s-group-structure: s-GroupStructure,
record-update: r[x := v],
record: record(x.T[x]),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
top: Top,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
ss-point: Point(ss),
ss-sep: x # y,
s-group-axioms: s-group-axioms(sg),
sg-op: (x y),
ss-eq: x ≡ y,
sq_stable: SqStable(P),
squash: ↓T,
sg-id: 1,
sg-inv: x^-1,
cand: A c∧ B
Lemmas referenced :
subtype_rel_self,
record-select_wf,
top_wf,
istype-atom,
not_wf,
all_wf,
or_wf,
eq_atom_wf,
uiff_transitivity,
equal-wf-base,
bool_wf,
atom_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
rec_select_update_lemma,
istype-void,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
istype-assert,
sq_stable__uall,
ss-point_wf,
ss-eq_wf,
sq_stable__ss-eq,
sq_stable__and,
s-group-axioms_wf,
ss-sep_wf,
separation-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesisEquality,
sqequalHypSubstitution,
dependentIntersectionElimination,
sqequalRule,
dependentIntersectionEqElimination,
thin,
hypothesis,
applyEquality,
tokenEquality,
instantiate,
extract_by_obid,
isectElimination,
universeEquality,
setEquality,
functionEquality,
cumulativity,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
because_Cache,
applyLambdaEquality,
setElimination,
rename,
inhabitedIsType,
universeIsType,
dependent_set_memberEquality_alt,
dependentIntersection_memberEquality,
functionExtensionality_alt,
lambdaFormation_alt,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
atomEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIstype,
sqequalBase,
functionIsType,
functionExtensionality,
isectEquality,
functionIsTypeImplies,
isectIsTypeImplies,
imageMemberEquality,
imageElimination,
isectIsType,
axiomEquality,
unionIsType,
setIsType,
productIsType
Latex:
\mforall{}[ss:SeparationSpace]. \mforall{}[e:Point(ss)]. \mforall{}[i:Point(ss) {}\mrightarrow{} Point(ss)]. \mforall{}[o:\{f:Point(ss)
{}\mrightarrow{} Point(ss)
{}\mrightarrow{} Point(ss)|
(\mforall{}x,y,z:Point(ss).
f x (f y z) \mequiv{} f (f x y)
z)
\mwedge{} (\mforall{}x:Point(ss). f x e \mequiv{} x)
\mwedge{} (\mforall{}x:Point(ss)
f x (i x) \mequiv{} e)\} ].
\mforall{}[sep:\mforall{}x,x',y,y':Point(ss). (o x y \# o x' y' {}\mRightarrow{} (x \# x' \mvee{} y \# y'))]. \mforall{}[invsep:\mforall{}x,y:Point(ss).
(i x \# i y
{}\mRightarrow{} x \# y)].
(mk-s-group(ss; e; i; o; sep; invsep) \mmember{} s-Group)
Date html generated:
2019_10_31-AM-07_27_49
Last ObjectModification:
2019_09_19-PM-04_29_43
Theory : constructive!algebra
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