Nuprl Lemma : grp_lt_op_l
∀[g:OGrp]. ∀[a,b,c:|g|]. uiff(a < b;(c * a) < (c * b))
Proof
Definitions occuring in Statement :
ocgrp: OGrp,
grp_lt: 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_lt: a < b,
set_lt: a <p b,
ocgrp: OGrp,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
subtype_rel: A ⊆r B,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
implies: P ⇒ Q,
prop: ℙ,
infix_ap: x f y,
squash: ↓T,
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_lt,
ocgrp_wf,
grp_lt_wf,
grp_op_wf,
set_car_wf,
grp_car_wf,
infix_ap_wf,
oset_of_ocmon_wf0,
set_blt_wf,
assert_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
lemma_by_obid,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
applyEquality,
lambdaEquality,
functionEquality,
independent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
imageElimination,
instantiate,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[g:OGrp]. \mforall{}[a,b,c:|g|]. uiff(a < b;(c * a) < (c * b))
Date html generated:
2016_05_15-PM-00_13_22
Last ObjectModification:
2016_01_15-PM-11_05_17
Theory : groups_1
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