Nuprl Lemma : oal_grp_wf2
∀s:LOSet. ∀g:OGrp.  (oal_grp(s;g) ∈ OGrp)
Proof
Definitions occuring in Statement : 
oal_grp: oal_grp(s;g)
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
ocgrp: OGrp
, 
loset: LOSet
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
ocgrp: OGrp
, 
uall: ∀[x:A]. B[x]
, 
ocmon: OCMon
, 
abmonoid: AbMon
, 
mon: Mon
, 
prop: ℙ
, 
squash: ↓T
, 
omon: OMon
, 
and: P ∧ Q
, 
grp: Group{i}
, 
abgrp: AbGrp
, 
guard: {T}
, 
uimplies: b supposing a
, 
cancel: Cancel(T;S;op)
, 
dmon: DMon
, 
abdmonoid: AbDMon
, 
grp_leq: a ≤ b
, 
dset: DSet
, 
dset_of_mon: g↓set
, 
set_prod: s × t
, 
dset_list: s List
, 
set_car: |p|
, 
mk_dset: mk_dset(T, eq)
, 
dset_set: dset_set, 
oalist: oal(a;b)
, 
implies: P 
⇒ Q
, 
infix_ap: x f y
, 
pi2: snd(t)
, 
grp_op: *
, 
pi1: fst(t)
, 
grp_car: |g|
, 
oal_grp: oal_grp(s;g)
, 
monot: monot(T;x,y.R[x; y];f)
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
bfalse: ff
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
band: p ∧b q
, 
btrue: tt
, 
it: ⋅
, 
unit: Unit
, 
bool: 𝔹
, 
so_apply: x[s1;s2]
, 
so_lambda: λ2x y.t[x; y]
Lemmas referenced : 
ocgrp_wf, 
loset_wf, 
oal_grp_wf, 
ocgrp_subtype_abdgrp, 
abdgrp_subtype_abgrp, 
oal_grp_wf1, 
inverse_wf, 
grp_car_wf, 
grp_op_wf, 
grp_id_wf, 
grp_inv_wf, 
infix_ap_wf, 
equal_wf, 
igrp_wf, 
grp_wf, 
abgrp_wf, 
subtype_rel_transitivity, 
abgrp_subtype_grp, 
grp_subtype_igrp, 
grp_op_cancel_l, 
comm_wf, 
dmon_wf, 
abdmonoid_wf, 
ocmon_wf, 
ocgrp_subtype_ocmon, 
ocmon_subtype_abdmonoid, 
abdmonoid_dmon, 
abmonoid_properties, 
ocmon_properties, 
ocgrp_properties, 
omon_lt_mono_imp_leq_mono, 
dset_wf, 
oalist_wf, 
set_car_wf, 
grp_lt_wf, 
oal_merge_wf2, 
oal_lt_iff_grp_lt, 
oal_merge_preserves_lt, 
monot_wf, 
uall_wf, 
cancel_wf, 
eqtt_to_assert, 
grp_eq_wf, 
grp_le_wf, 
bool_wf, 
assert_wf, 
ulinorder_wf, 
grp_properties, 
abgrp_properties
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
sqequalHypSubstitution, 
hypothesis, 
introduction, 
extract_by_obid, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
applyEquality, 
sqequalRule, 
dependent_set_memberEquality, 
isectElimination, 
setElimination, 
rename, 
because_Cache, 
imageElimination, 
baseClosed, 
imageMemberEquality, 
applyLambdaEquality, 
equalitySymmetry, 
equalityTransitivity, 
independent_pairFormation, 
axiomEquality, 
isect_memberEquality, 
lambdaEquality, 
independent_isectElimination, 
instantiate, 
productElimination, 
isect_memberFormation, 
independent_functionElimination, 
equalityElimination, 
unionElimination, 
functionEquality, 
productEquality
Latex:
\mforall{}s:LOSet.  \mforall{}g:OGrp.    (oal\_grp(s;g)  \mmember{}  OGrp)
Date html generated:
2019_10_16-PM-01_08_49
Last ObjectModification:
2018_08_22-AM-11_53_49
Theory : polynom_2
Home
Index