Nuprl Lemma : oal_neg_eq_nil
∀a:LOSet. ∀b:AbDGrp. ∀ps:|oal(a;b)|. ((--ps) = 00 ∈ |oal(a;b)| ⇐⇒ ps = 00 ∈ |oal(a;b)|)
Proof
Definitions occuring in Statement :
oal_neg: --ps,
oal_nil: 00,
oalist: oal(a;b),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T,
abdgrp: AbDGrp,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
abdgrp: AbDGrp,
abgrp: AbGrp,
grp: Group{i},
abdmonoid: AbDMon,
dmon: DMon,
uall: ∀[x:A]. B[x],
mon: Mon,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
rev_implies: P ⇐ Q,
dset: DSet,
true: True,
guard: {T}
Lemmas referenced :
oal_neg_wf2,
oal_nil_wf,
subtype_rel_sets,
mon_wf,
inverse_wf,
grp_car_wf,
grp_op_wf,
grp_id_wf,
grp_inv_wf,
comm_wf,
eqfun_p_wf,
grp_eq_wf,
sq_stable__comm,
set_car_wf,
oalist_wf,
abdgrp_wf,
loset_wf,
oal_merge_wf2,
equal_wf,
squash_wf,
true_wf,
istype-universe,
oal_neg_left_inv,
oal_nil_ident_l,
subtype_rel_self,
iff_weakening_equal,
oal_neg_right_inv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
independent_pairFormation,
equalityIstype,
because_Cache,
applyEquality,
sqequalRule,
instantiate,
isectElimination,
setEquality,
cumulativity,
setElimination,
rename,
lambdaEquality_alt,
setIsType,
universeIsType,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
applyLambdaEquality,
universeEquality,
natural_numberEquality,
productElimination
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDGrp. \mforall{}ps:|oal(a;b)|. ((--ps) = 00 \mLeftarrow{}{}\mRightarrow{} ps = 00)
Date html generated:
2019_10_16-PM-01_07_49
Last ObjectModification:
2018_11_27-AM-10_45_26
Theory : polynom_2
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