Nuprl Lemma : rng_times_nat_op_r

`∀[r:Rng]. ∀[a,b:|r|]. ∀[n:ℕ].  (((n ⋅r b) * a) = (n ⋅r (b * a)) ∈ |r|)`

Proof

Definitions occuring in Statement :  rng_nat_op: `n ⋅r e` rng: `Rng` rng_times: `*` rng_car: `|r|` nat: `ℕ` uall: `∀[x:A]. B[x]` infix_ap: `x f y` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` rng_nat_op: `n ⋅r e` mon_nat_op: `n ⋅ e` nat_op: `n x(op;id) e` mon_itop: `Π lb ≤ i < ub. E[i]` rng_sum: rng_sum rng: `Rng` nat: `ℕ` uimplies: `b supposing a` ge: `i ≥ j ` 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: `P `` Q` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  int_seg_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties rng_times_sum_r rng_wf rng_car_wf nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis lemma_by_obid sqequalRule sqequalHypSubstitution isect_memberEquality isectElimination thin hypothesisEquality axiomEquality because_Cache setElimination rename natural_numberEquality independent_isectElimination dependent_functionElimination unionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll

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

Date html generated: 2016_05_15-PM-00_28_30
Last ObjectModification: 2016_01_15-AM-08_51_25

Theory : rings_1

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