Nuprl Lemma : oalist_hgrp_eqs
∀s:LOSet. ∀g:OGrp. ∀a1,a2:|oal(s;g↓hgrp)|. ((a1 = a2 ∈ |oal(s;g)|) ⇒ (a1 = a2 ∈ |oal(s;g↓hgrp)|))
Proof
Definitions occuring in Statement :
oalist: oal(a;b),
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T,
hgrp_of_ocgrp: g↓hgrp,
ocgrp: OGrp,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
grp_id: e,
pi1: fst(t),
pi2: snd(t),
hgrp_of_ocgrp: g↓hgrp,
grp_car: |g|,
hgrp_car: |g|+,
ocgrp: OGrp,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x.t[x],
so_apply: x[s],
dset: DSet
Lemmas referenced :
equal_wf,
set_car_wf,
oalist_wf,
ocmon_subtype_abdmonoid,
ocgrp_subtype_ocmon,
subtype_rel_transitivity,
ocgrp_wf,
ocmon_wf,
abdmonoid_wf,
set_car_inc,
hgrp_of_ocgrp_wf2,
loset_wf,
oalist_strong-subtype,
grp_id_wf,
hgrp_of_ocgrp_wf,
grp_car_wf,
grp_leq_wf,
strong-subtype-self,
strong-subtype-set3,
strong-subtype-implies,
dset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
independent_pairFormation,
lambdaEquality,
setElimination,
rename,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}s:LOSet. \mforall{}g:OGrp. \mforall{}a1,a2:|oal(s;g\mdownarrow{}hgrp)|. ((a1 = a2) {}\mRightarrow{} (a1 = a2))
Date html generated:
2016_05_16-AM-08_22_29
Last ObjectModification:
2015_12_28-PM-06_26_42
Theory : polynom_2
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