Nuprl Lemma : rng_nat_op_add
∀[r:Rng]. ∀[e:|r|]. ∀[a,b:ℕ]. (((a + b) ⋅r e) = ((a ⋅r e) +r (b ⋅r e)) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_nat_op: n ⋅r e,
rng: Rng,
rng_plus: +r,
rng_car: |r|,
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,
subtype_rel: A ⊆r B,
grp: Group{i},
mon: Mon,
imon: IMonoid,
prop: ℙ,
rng_nat_op: n ⋅r e,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t),
rng: Rng
Lemmas referenced :
mon_nat_op_add,
add_grp_of_rng_wf_a,
grp_sig_wf,
monoid_p_wf,
grp_car_wf,
grp_op_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
nat_wf,
rng_car_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
cumulativity,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[r:Rng]. \mforall{}[e:|r|]. \mforall{}[a,b:\mBbbN{}]. (((a + b) \mcdot{}r e) = ((a \mcdot{}r e) +r (b \mcdot{}r e)))
Date html generated:
2016_05_15-PM-00_27_25
Last ObjectModification:
2015_12_26-PM-11_58_54
Theory : rings_1
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