Nuprl Lemma : grp_leq_shift_right
∀[g:OGrp]. ∀[a,b:|g|]. uiff(a ≤ b;e ≤ (b * (~ a)))
Proof
Definitions occuring in Statement :
ocgrp: OGrp,
grp_leq: a ≤ b,
grp_inv: ~,
grp_id: e,
grp_op: *,
grp_car: |g|,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
infix_ap: x f y,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
grp_leq: a ≤ b,
ocgrp: OGrp,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
implies: P ⇒ Q,
prop: ℙ,
infix_ap: x f y,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
grp_leq_op_l,
iff_weakening_uiff,
uiff_wf,
iabmonoid_wf,
abmonoid_wf,
abdmonoid_wf,
ocmon_wf,
ocgrp_subtype_ocmon,
ocmon_subtype_abdmonoid,
abdmonoid_abmonoid,
abmonoid_subtype_iabmonoid,
abmonoid_comm,
iff_weakening_equal,
igrp_wf,
grp_wf,
abgrp_wf,
subtype_rel_transitivity,
ocgrp_subtype_abgrp,
abgrp_subtype_grp,
grp_subtype_igrp,
grp_inverse,
grp_sig_wf,
true_wf,
squash_wf,
ocgrp_wf,
grp_leq_wf,
grp_inv_wf,
grp_op_wf,
grp_id_wf,
grp_le_wf,
bool_wf,
grp_car_wf,
infix_ap_wf,
assert_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
isect_memberEquality,
isectElimination,
hypothesisEquality,
lemma_by_obid,
setElimination,
rename,
hypothesis,
because_Cache,
applyEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
addLevel,
independent_isectElimination,
lambdaEquality,
imageElimination,
instantiate,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
cumulativity
Latex:
\mforall{}[g:OGrp]. \mforall{}[a,b:|g|]. uiff(a \mleq{} b;e \mleq{} (b * (\msim{} a)))
Date html generated:
2016_05_15-PM-00_13_41
Last ObjectModification:
2016_01_15-PM-11_05_56
Theory : groups_1
Home
Index