Nuprl Lemma : mon_nat_op_unroll
∀[g:IMonoid]. ∀[n:ℕ+]. ∀[e:|g|]. ((n ⋅ e) = (((n - 1) ⋅ e) * e) ∈ |g|)
Proof
Definitions occuring in Statement :
mon_nat_op: n ⋅ e,
imon: IMonoid,
grp_op: *,
grp_car: |g|,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
mon_nat_op: n ⋅ e,
nat_op: n x(op;id) e,
imon: IMonoid,
nat_plus: ℕ+,
uimplies: b supposing a,
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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
int_seg_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
itop_unroll_hi,
imon_wf,
nat_plus_wf,
grp_car_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
natural_numberEquality,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[g:IMonoid]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[e:|g|]. ((n \mcdot{} e) = (((n - 1) \mcdot{} e) * e))
Date html generated:
2016_05_15-PM-00_16_37
Last ObjectModification:
2016_01_15-PM-11_04_00
Theory : groups_1
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