Nuprl Lemma : mon_nat_op_hom_swap
∀[g,h:IMonoid]. ∀[f:MonHom(g,h)]. ∀[n:ℕ]. ∀[u:|g|]. ((n ⋅ (f u)) = (f (n ⋅ u)) ∈ |h|)
Proof
Definitions occuring in Statement :
mon_nat_op: n ⋅ e,
monoid_hom: MonHom(M1,M2),
imon: IMonoid,
grp_car: |g|,
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
imon: IMonoid,
monoid_hom_p: IsMonHom{M1,M2}(f),
and: P ∧ Q,
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
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,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
monoid_hom: MonHom(M1,M2),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
infix_ap: x f y
Lemmas referenced :
grp_car_wf,
nat_wf,
monoid_hom_wf,
imon_wf,
monoid_hom_properties,
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,
iff_weakening_equal,
mon_nat_op_unroll,
grp_op_wf,
mon_nat_op_wf,
le_wf,
infix_ap_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
productElimination,
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,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality
Latex:
\mforall{}[g,h:IMonoid]. \mforall{}[f:MonHom(g,h)]. \mforall{}[n:\mBbbN{}]. \mforall{}[u:|g|]. ((n \mcdot{} (f u)) = (f (n \mcdot{} u)))
Date html generated:
2017_10_01-AM-08_16_39
Last ObjectModification:
2017_02_28-PM-02_01_59
Theory : groups_1
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