Nuprl Lemma : grp_op_polarity
∀[g:OGrp]. ∀[a,b:|g|].  (e ≤ (a * b)) supposing ((e ≤ b) and (e ≤ a))
Proof
Definitions occuring in Statement : 
ocgrp: OGrp
, 
grp_leq: a ≤ b
, 
grp_id: e
, 
grp_op: *
, 
grp_car: |g|
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
infix_ap: x f y
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
grp_leq: a ≤ b
, 
infix_ap: x f y
, 
ocgrp: OGrp
, 
ocmon: OCMon
, 
abmonoid: AbMon
, 
mon: Mon
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
squash: ↓T
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
true: True
, 
iff: P 
⇐⇒ Q
Lemmas referenced : 
grp_leq_transitivity, 
iff_weakening_equal, 
imon_wf, 
iabmonoid_wf, 
abmonoid_wf, 
abdmonoid_wf, 
ocmon_wf, 
subtype_rel_transitivity, 
ocgrp_subtype_ocmon, 
ocmon_subtype_abdmonoid, 
abdmonoid_abmonoid, 
abmonoid_subtype_iabmonoid, 
iabmonoid_subtype_imon, 
mon_ident, 
grp_sig_wf, 
true_wf, 
squash_wf, 
grp_leq_op_l, 
ocgrp_wf, 
grp_car_wf, 
grp_leq_wf, 
grp_op_wf, 
grp_id_wf, 
grp_le_wf, 
assert_witness
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
sqequalRule, 
sqequalHypSubstitution, 
lemma_by_obid, 
isectElimination, 
thin, 
applyEquality, 
setElimination, 
rename, 
hypothesisEquality, 
hypothesis, 
independent_functionElimination, 
isect_memberEquality, 
because_Cache, 
equalityTransitivity, 
equalitySymmetry, 
productElimination, 
independent_isectElimination, 
lambdaEquality, 
imageElimination, 
instantiate, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
universeEquality
Latex:
\mforall{}[g:OGrp].  \mforall{}[a,b:|g|].    (e  \mleq{}  (a  *  b))  supposing  ((e  \mleq{}  b)  and  (e  \mleq{}  a))
Date html generated:
2016_05_15-PM-00_13_26
Last ObjectModification:
2016_01_15-PM-11_05_14
Theory : groups_1
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