Nuprl Lemma : grp_leq_op_l
∀[g:OGrp]. ∀[a,b,c:|g|]. uiff(a ≤ b;(c * a) ≤ (c * b))
Proof
Definitions occuring in Statement :
ocgrp: OGrp,
grp_leq: a ≤ b,
grp_op: *,
grp_car: |g|,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
infix_ap: x f y
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,
infix_ap: x f y,
ocgrp: OGrp,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
implies: P ⇒ Q,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
true: True,
iff: P ⇐⇒ Q
Lemmas referenced :
iff_weakening_equal,
igrp_wf,
grp_wf,
abgrp_wf,
subtype_rel_transitivity,
ocgrp_subtype_abgrp,
abgrp_subtype_grp,
grp_subtype_igrp,
grp_inv_assoc,
grp_sig_wf,
true_wf,
squash_wf,
grp_inv_wf,
grp_op_preserves_le,
ocgrp_wf,
grp_car_wf,
grp_leq_wf,
grp_op_wf,
grp_le_wf,
assert_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
lemma_by_obid,
isectElimination,
thin,
applyEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality,
imageElimination,
instantiate,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[g:OGrp]. \mforall{}[a,b,c:|g|]. uiff(a \mleq{} b;(c * a) \mleq{} (c * b))
Date html generated:
2016_05_15-PM-00_13_19
Last ObjectModification:
2016_01_15-PM-11_05_24
Theory : groups_1
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