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
Home
Index