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: 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,
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,
and: P ∧ Q,
prop: ℙ,
iabmonoid: IAbMonoid,
imon: IMonoid,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
infix_ap: x f y,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+
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
Home
Index