Nuprl Lemma : mon_itop_split_el

[g:IMonoid]. ∀[a,b,c:ℤ].
  (∀[E:{a..c-} ⟶ |g|]
     ((Π a ≤ j < c. E[j]) ((Π a ≤ j < b. E[j]) (E[b] (Π 1 ≤ j < c. E[j]))) ∈ |g|)) supposing 
     (b < and 
     (a ≤ b))


Proof




Definitions occuring in Statement :  mon_itop: Π lb ≤ i < ub. E[i] imon: IMonoid grp_op: * grp_car: |g| int_seg: {i..j-} less_than: a < b uimplies: supposing a uall: [x:A]. B[x] infix_ap: y so_apply: x[s] le: A ≤ B function: x:A ⟶ B[x] add: m natural_number: $n int: equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a imon: IMonoid prop: squash: T all: x:A. B[x] decidable: Dec(P) or: P ∨ Q satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False implies:  Q not: ¬A top: Top and: P ∧ Q so_lambda: λ2x.t[x] so_apply: x[s] int_seg: {i..j-} lelt: i ≤ j < k true: True subtype_rel: A ⊆B guard: {T} iff: ⇐⇒ Q rev_implies:  Q infix_ap: y
Lemmas referenced :  int_seg_wf grp_car_wf less_than_wf le_wf imon_wf equal_wf squash_wf true_wf mon_itop_split decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf infix_ap_wf grp_op_wf mon_itop_wf decidable__lt lelt_wf itermAdd_wf itermConstant_wf int_term_value_add_lemma int_term_value_constant_lemma iff_weakening_equal mon_itop_unroll_lo
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis functionEquality extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality setElimination rename sqequalRule isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry intEquality applyEquality lambdaEquality imageElimination universeEquality independent_isectElimination dependent_functionElimination unionElimination natural_numberEquality dependent_pairFormation int_eqEquality voidElimination voidEquality independent_pairFormation computeAll functionExtensionality dependent_set_memberEquality productElimination addEquality imageMemberEquality baseClosed independent_functionElimination

Latex:
\mforall{}[g:IMonoid].  \mforall{}[a,b,c:\mBbbZ{}].
    (\mforall{}[E:\{a..c\msupminus{}\}  {}\mrightarrow{}  |g|]
          ((\mPi{}  a  \mleq{}  j  <  c.  E[j])  =  ((\mPi{}  a  \mleq{}  j  <  b.  E[j])  *  (E[b]  *  (\mPi{}  b  +  1  \mleq{}  j  <  c.  E[j])))))  supposing 
          (b  <  c  and 
          (a  \mleq{}  b))



Date html generated: 2017_10_01-AM-08_16_17
Last ObjectModification: 2017_02_28-PM-02_01_17

Theory : groups_1


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