Nuprl Lemma : oal_lv_neg
∀s:LOSet. ∀g:AbDGrp. ∀ps:|oal(s;g)|. ((¬(ps = 00 ∈ |oal(s;g)|)) ⇒ (lv(--ps) = (~ lv(ps)) ∈ |g|))
Proof
Definitions occuring in Statement :
oal_lv: lv(ps),
oal_neg: --ps,
oal_nil: 00,
oalist: oal(a;b),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
equal: s = t ∈ T,
abdgrp: AbDGrp,
grp_inv: ~,
grp_car: |g|,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
abdgrp: AbDGrp,
abgrp: AbGrp,
grp: Group{i},
mon: Mon,
subtype_rel: A ⊆r B,
abdmonoid: AbDMon,
dmon: DMon,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
sq_stable: SqStable(P),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
dset: DSet,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
dset_list: s List,
set_prod: s × t,
dset_of_mon: g↓set
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
grp_car_wf,
lookup_oal_lk,
subtype_rel_sets,
mon_wf,
inverse_wf,
grp_op_wf,
grp_id_wf,
grp_inv_wf,
comm_wf,
eqfun_p_wf,
grp_eq_wf,
sq_stable__comm,
oal_neg_wf2,
subtype_rel_self,
iff_weakening_equal,
oal_nil_wf,
istype-void,
set_car_wf,
oalist_wf,
abdgrp_wf,
loset_wf,
oal_merge_wf2,
oal_neg_left_inv,
oal_nil_ident_l,
lookup_wf,
list_wf,
poset_sig_wf,
oal_lk_neg,
oal_neg_wf,
oal_lk_wf,
lookup_oal_neg,
grp_subtype_igrp,
abgrp_subtype_grp,
abdgrp_subtype_abgrp,
subtype_rel_transitivity,
abgrp_wf,
grp_wf,
igrp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
inhabitedIsType,
instantiate,
universeEquality,
setElimination,
rename,
dependent_functionElimination,
sqequalRule,
setEquality,
cumulativity,
because_Cache,
setIsType,
independent_isectElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
natural_numberEquality,
productElimination,
functionIsType,
equalityIstype,
applyLambdaEquality,
voidElimination,
productEquality
Latex:
\mforall{}s:LOSet. \mforall{}g:AbDGrp. \mforall{}ps:|oal(s;g)|. ((\mneg{}(ps = 00)) {}\mRightarrow{} (lv(--ps) = (\msim{} lv(ps))))
Date html generated:
2019_10_16-PM-01_08_22
Last ObjectModification:
2018_11_27-AM-10_31_05
Theory : polynom_2
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