### Nuprl Lemma : rng_all_properties

`∀[r:Rng]. (Assoc(|r|;+r) ∧ Ident(|r|;+r;0) ∧ Inverse(|r|;+r;0;-r) ∧ Assoc(|r|;*) ∧ Ident(|r|;*;1) ∧ BiLinear(|r|;+r;*))`

Proof

Definitions occuring in Statement :  rng: `Rng` rng_one: `1` rng_times: `*` rng_minus: `-r` rng_zero: `0` rng_plus: `+r` rng_car: `|r|` bilinear: `BiLinear(T;pl;tm)` ident: `Ident(T;op;id)` inverse: `Inverse(T;op;id;inv)` assoc: `Assoc(T;op)` uall: `∀[x:A]. B[x]` and: `P ∧ Q`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` rng: `Rng` and: `P ∧ Q` assoc: `Assoc(T;op)` ident: `Ident(T;op;id)` inverse: `Inverse(T;op;id;inv)` bilinear: `BiLinear(T;pl;tm)` ring_p: `IsRing(T;plus;zero;neg;times;one)` monoid_p: `IsMonoid(T;op;id)` group_p: `IsGroup(T;op;id;inv)` cand: `A c∧ B`
Lemmas referenced :  rng_properties 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 setElimination rename sqequalRule productElimination independent_pairEquality isect_memberEquality axiomEquality independent_pairFormation

Latex:
\mforall{}[r:Rng]
(Assoc(|r|;+r)
\mwedge{}  Ident(|r|;+r;0)
\mwedge{}  Inverse(|r|;+r;0;-r)
\mwedge{}  Assoc(|r|;*)
\mwedge{}  Ident(|r|;*;1)
\mwedge{}  BiLinear(|r|;+r;*))

Date html generated: 2016_05_15-PM-00_20_35
Last ObjectModification: 2015_12_27-AM-00_02_49

Theory : rings_1

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