Nuprl Lemma : grp_lt_is_sp_of_leq_a
∀[g:OMon]. ∀[a,b:|g|]. uiff(a < b;(a ≤ b) ∧ (¬(b ≤ a)))
Proof
Definitions occuring in Statement :
grp_lt: a < b,
grp_leq: a ≤ b,
omon: OMon,
grp_car: |g|,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
omon: OMon,
abmonoid: AbMon,
mon: Mon,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
set_leq: a ≤ b,
set_le: ≤b,
pi2: snd(t),
grp_lt: a < b,
grp_leq: a ≤ b,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
infix_ap: x f y,
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
set_lt: a <p b
Lemmas referenced :
set_lt_is_sp_of_leq_a,
oset_of_ocmon_wf0,
assert_witness,
grp_le_wf,
grp_leq_wf,
grp_lt_wf,
set_blt_wf,
and_wf,
not_wf,
grp_car_wf,
omon_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
isect_memberEquality,
productElimination,
independent_pairEquality,
applyEquality,
independent_functionElimination,
lambdaEquality,
dependent_functionElimination,
voidElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[g:OMon]. \mforall{}[a,b:|g|]. uiff(a < b;(a \mleq{} b) \mwedge{} (\mneg{}(b \mleq{} a)))
Date html generated:
2016_05_15-PM-00_12_02
Last ObjectModification:
2015_12_26-PM-11_43_03
Theory : groups_1
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