Nuprl Lemma : nat_op_on_nat_add_mon
∀[m,n:ℕ]. ((m ⋅ n) = (m * n) ∈ ℕ)
Proof
Definitions occuring in Statement :
nat_add_mon: <ℕ,+>,
mon_nat_op: n ⋅ e,
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,
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,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
nat_add_mon: <ℕ,+>,
grp_car: |g|,
pi1: fst(t),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
grp_id: e,
pi2: snd(t),
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
infix_ap: x f y,
grp_op: *
Lemmas referenced :
nat_wf,
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,
nat_add_mon_wf2,
iabmonoid_subtype_imon,
abmonoid_subtype_iabmonoid,
abdmonoid_abmonoid,
ocmon_subtype_abdmonoid,
subtype_rel_transitivity,
ocmon_wf,
abdmonoid_wf,
abmonoid_wf,
iabmonoid_wf,
imon_wf,
grp_car_wf,
nat_add_mon_wf,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
le_wf,
iff_weakening_equal,
false_wf,
mon_nat_op_unroll,
mul_bounds_1a,
grp_op_wf,
itermAdd_wf,
int_term_value_add_lemma
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,
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,
instantiate,
multiplyEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
productElimination,
applyLambdaEquality
Latex:
\mforall{}[m,n:\mBbbN{}]. ((m \mcdot{} n) = (m * n))
Date html generated:
2017_10_01-AM-08_17_01
Last ObjectModification:
2017_02_28-PM-02_02_28
Theory : groups_1
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