Nuprl Lemma : mon_nat_op_mul
∀[g:IMonoid]. ∀[m,n:ℕ]. ∀[e:|g|]. ((n ⋅ (m ⋅ e)) = ((n * m) ⋅ e) ∈ |g|)
Proof
Definitions occuring in Statement :
mon_nat_op: n ⋅ e,
imon: IMonoid,
grp_car: |g|,
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
imon: IMonoid,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
infix_ap: x f y,
subtract: n - m
Lemmas referenced :
istype-nat,
imon_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
zero-mul,
equal_wf,
istype-universe,
grp_car_wf,
mon_nat_op_wf,
istype-false,
istype-le,
mon_nat_op_zero,
iff_weakening_equal,
subtract-1-ge-0,
squash_wf,
true_wf,
mon_nat_op_unroll,
mul_bounds_1a,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
subtype_rel_self,
grp_op_wf,
mon_nat_op_add,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
mul-distributes-right,
add-associates,
mul-commutes,
add-swap,
add-commutes,
add-mul-special,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
isectIsTypeImplies,
inhabitedIsType,
extract_by_obid,
universeIsType,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
voidElimination,
independent_pairFormation,
functionIsTypeImplies,
applyEquality,
imageElimination,
because_Cache,
instantiate,
dependent_set_memberEquality_alt,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
universeEquality,
multiplyEquality,
unionElimination,
minusEquality
Latex:
\mforall{}[g:IMonoid]. \mforall{}[m,n:\mBbbN{}]. \mforall{}[e:|g|]. ((n \mcdot{} (m \mcdot{} e)) = ((n * m) \mcdot{} e))
Date html generated:
2019_10_15-AM-10_33_04
Last ObjectModification:
2018_10_19-AM-08_57_20
Theory : groups_1
Home
Index