Nuprl Lemma : oal_lt_char
∀s:LOSet. ∀g:OGrp. ((ps,qs:|oal(s;g)|. ps << qs) <≡>{|oal(s;g)|} ((ps,qs:|oal(s;g)|. ps ≤{s,g} qs)\))
Proof
Definitions occuring in Statement :
oal_lt: ps << qs,
oal_le: ps ≤{s,g} qs,
oalist: oal(a;b),
binrel_eqv: E <≡>{T} E',
all: ∀x:A. B[x],
ocgrp: OGrp,
loset: LOSet,
set_car: |p|,
s_part: E\,
ab_binrel: x,y:T. E[x; y]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
ocgrp: OGrp,
so_apply: x[s1;s2],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
xxirrefl: irrefl(T;R),
ab_binrel: x,y:T. E[x; y],
xxtrans: trans(T;E)
Lemmas referenced :
ocgrp_wf,
loset_wf,
binrel_eqv_functionality_wrt_breqv,
set_car_wf,
oalist_wf,
ocmon_subtype_abdmonoid,
ocgrp_subtype_ocmon,
subtype_rel_transitivity,
ocmon_wf,
abdmonoid_wf,
ab_binrel_wf,
oal_lt_wf,
s_part_wf,
refl_cl_wf,
binrel_eqv_weakening,
sp_refl_cl_cancel,
oal_le_wf,
ocgrp_subtype_abdgrp,
s_part_functionality_wrt_breqv,
oal_le_char,
binrel_eqv_wf,
oal_lt_irrefl,
irrefl_trans_imp_sasym,
oal_lt_trans
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
hypothesis,
addLevel,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
lambdaEquality,
setElimination,
rename,
independent_functionElimination,
productElimination,
cumulativity,
universeEquality
Latex:
\mforall{}s:LOSet. \mforall{}g:OGrp.
((ps,qs:|oal(s;g)|. ps << qs) <\mequiv{}>\{|oal(s;g)|\} ((ps,qs:|oal(s;g)|. ps \mleq{}\{s,g\} qs)\mbackslash{}))
Date html generated:
2016_05_16-AM-08_21_35
Last ObjectModification:
2015_12_28-PM-06_25_32
Theory : polynom_2
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