Nuprl Lemma : mon_nat_op_op

[g:IAbMonoid]. ∀[n:ℕ]. ∀[a,b:|g|].  ((n ⋅ (a b)) ((n ⋅ a) (n ⋅ b)) ∈ |g|)


Proof




Definitions occuring in Statement :  mon_nat_op: n ⋅ e iabmonoid: IAbMonoid grp_op: * grp_car: |g| nat: uall: [x:A]. B[x] infix_ap: y equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T nat: implies:  Q false: False ge: i ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A all: x:A. B[x] top: Top and: P ∧ Q prop: iabmonoid: IAbMonoid imon: IMonoid decidable: Dec(P) or: P ∨ Q squash: T infix_ap: y true: True subtype_rel: A ⊆B guard: {T} iff: ⇐⇒ Q rev_implies:  Q nat_plus: +
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf grp_car_wf decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf iabmonoid_wf equal_wf squash_wf true_wf mon_nat_op_zero grp_op_wf iff_weakening_equal grp_id_wf mon_ident mon_nat_op_unroll infix_ap_wf mon_nat_op_wf le_wf mon_assoc abmonoid_ac_1
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination axiomEquality because_Cache unionElimination applyEquality imageElimination equalityTransitivity equalitySymmetry universeEquality imageMemberEquality baseClosed productElimination dependent_set_memberEquality

Latex:
\mforall{}[g:IAbMonoid].  \mforall{}[n:\mBbbN{}].  \mforall{}[a,b:|g|].    ((n  \mcdot{}  (a  *  b))  =  ((n  \mcdot{}  a)  *  (n  \mcdot{}  b)))



Date html generated: 2017_10_01-AM-08_16_35
Last ObjectModification: 2017_02_28-PM-02_02_22

Theory : groups_1


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