Nuprl Lemma : grp_lt_transitivity_1

[g:OCMon]. ∀[a,b,c:|g|].  (a < c) supposing ((b < c) and (a ≤ b))


Definitions occuring in Statement :  grp_lt: a < b grp_leq: a ≤ b ocmon: OCMon grp_car: |g| uimplies: supposing a uall: [x:A]. B[x]
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T subtype_rel: A ⊆B ocmon: OCMon omon: OMon so_lambda: λ2x.t[x] prop: and: P ∧ Q abmonoid: AbMon mon: Mon so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] all: x:A. B[x] implies:  Q bool: 𝔹 unit: Unit it: btrue: tt band: p ∧b q ifthenelse: if then else fi  uiff: uiff(P;Q) uimplies: supposing a bfalse: ff infix_ap: y so_apply: x[s] cand: c∧ B oset_of_ocmon: g↓oset dset_of_mon: g↓set set_car: |p| pi1: fst(t) set_leq: a ≤ b set_le: b pi2: snd(t) grp_leq: a ≤ b grp_lt: a < b set_lt: a <b
Lemmas referenced :  set_lt_transitivity_1 oset_of_ocmon_wf subtype_rel_sets abmonoid_wf ulinorder_wf grp_car_wf assert_wf infix_ap_wf bool_wf grp_le_wf equal_wf grp_eq_wf eqtt_to_assert cancel_wf grp_op_wf uall_wf monot_wf assert_witness set_blt_wf oset_of_ocmon_wf0 grp_lt_wf grp_leq_wf ocmon_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality applyEquality sqequalRule instantiate hypothesis because_Cache lambdaEquality productEquality setElimination rename cumulativity universeEquality functionEquality lambdaFormation unionElimination equalityElimination productElimination independent_isectElimination equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination setEquality independent_pairFormation isect_memberEquality

\mforall{}[g:OCMon].  \mforall{}[a,b,c:|g|].    (a  <  c)  supposing  ((b  <  c)  and  (a  \mleq{}  b))

Date html generated: 2017_10_01-AM-08_14_46
Last ObjectModification: 2017_02_28-PM-01_59_36

Theory : groups_1

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