### Nuprl Lemma : mon_nat_op_hom_swap

`∀[g,h:IMonoid]. ∀[f:MonHom(g,h)]. ∀[n:ℕ]. ∀[u:|g|].  ((n ⋅ (f u)) = (f (n ⋅ u)) ∈ |h|)`

Proof

Definitions occuring in Statement :  mon_nat_op: `n ⋅ e` monoid_hom: `MonHom(M1,M2)` imon: `IMonoid` grp_car: `|g|` nat: `ℕ` uall: `∀[x:A]. B[x]` apply: `f a` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` imon: `IMonoid` monoid_hom_p: `IsMonHom{M1,M2}(f)` and: `P ∧ Q` fun_thru_2op: `FunThru2op(A;B;opa;opb;f)` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` all: `∀x:A. B[x]` top: `Top` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` squash: `↓T` monoid_hom: `MonHom(M1,M2)` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` nat_plus: `ℕ+` infix_ap: `x f y`
Lemmas referenced :  grp_car_wf nat_wf monoid_hom_wf imon_wf monoid_hom_properties 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 decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma equal_wf squash_wf true_wf mon_nat_op_zero iff_weakening_equal mon_nat_op_unroll grp_op_wf mon_nat_op_wf le_wf infix_ap_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality sqequalRule isect_memberEquality axiomEquality because_Cache productElimination intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination unionElimination applyEquality imageElimination equalityTransitivity equalitySymmetry universeEquality imageMemberEquality baseClosed dependent_set_memberEquality

Latex:
\mforall{}[g,h:IMonoid].  \mforall{}[f:MonHom(g,h)].  \mforall{}[n:\mBbbN{}].  \mforall{}[u:|g|].    ((n  \mcdot{}  (f  u))  =  (f  (n  \mcdot{}  u)))

Date html generated: 2017_10_01-AM-08_16_39
Last ObjectModification: 2017_02_28-PM-02_01_59

Theory : groups_1

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