Nuprl Lemma : nat_op_add
∀[g:IMonoid]. ∀[e:|g|]. ∀[a,b:ℕ]. (a + b x(*;e) e = (a x(*;e) e * b x(*;e) e) ∈ |g|)
Proof
Definitions occuring in Statement :
nat_op: n x(op;id) e,
imon: IMonoid,
grp_id: e,
grp_op: *,
grp_car: |g|,
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
add: n + m,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
imon: IMonoid,
nat_op: n x(op;id) e,
squash: ↓T,
prop: ℙ,
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y
Lemmas referenced :
nat_wf,
grp_car_wf,
imon_wf,
equal_wf,
squash_wf,
true_wf,
itop_split,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
itermAdd_wf,
int_term_value_add_lemma,
int_seg_wf,
infix_ap_wf,
grp_op_wf,
itop_wf,
grp_id_wf,
iff_weakening_equal,
itop_shift,
minus-one-mul,
add-mul-special,
zero-mul,
add-associates,
add-commutes,
minus-one-mul-top,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
setElimination,
rename,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
addEquality,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
minusEquality,
functionEquality,
cumulativity
Latex:
\mforall{}[g:IMonoid]. \mforall{}[e:|g|]. \mforall{}[a,b:\mBbbN{}]. (a + b x(*;e) e = (a x(*;e) e * b x(*;e) e))
Date html generated:
2017_10_01-AM-08_16_04
Last ObjectModification:
2017_02_28-PM-02_01_04
Theory : groups_1
Home
Index