### Nuprl Lemma : quot_ring_car_elim_a

`∀r:CRng. ∀a:Ideal(r){i}. ∀d:detach_fun(|r|;a).`
`  ((∀w:|r|. SqStable(a w)) `` (∀u,v:|r|.  (u = v ∈ |r / d| `⇐⇒` a (u +r (-r v)))))`

Proof

Definitions occuring in Statement :  quot_ring: `r / d` ideal: `Ideal(r){i}` crng: `CRng` rng_minus: `-r` rng_plus: `+r` rng_car: `|r|` detach_fun: `detach_fun(T;A)` sq_stable: `SqStable(P)` infix_ap: `x f y` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` apply: `f a` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` crng: `CRng` rng: `Rng` prop: `ℙ` so_lambda: `λ2x.t[x]` ideal: `Ideal(r){i}` so_apply: `x[s]` quot_ring: `r / d` rng_car: `|r|` pi1: `fst(t)` iff: `P `⇐⇒` Q` and: `P ∧ Q` uiff: `uiff(P;Q)` uimplies: `b supposing a` subtype_rel: `A ⊆r B` rev_implies: `P `` Q` detach_fun: `detach_fun(T;A)` infix_ap: `x f y`
Lemmas referenced :  rng_car_wf all_wf sq_stable_wf detach_fun_wf ideal_wf crng_wf quot_ring_car_elim equal_wf quot_ring_car_wf quot_ring_car_subtype iff_wf assert_wf rng_plus_wf rng_minus_wf det_ideal_ap_elim
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis sqequalRule lambdaEquality applyEquality because_Cache addLevel productElimination independent_pairFormation impliesFunctionality independent_functionElimination independent_isectElimination dependent_functionElimination

Latex:
\mforall{}r:CRng.  \mforall{}a:Ideal(r)\{i\}.  \mforall{}d:detach\_fun(|r|;a).
((\mforall{}w:|r|.  SqStable(a  w))  {}\mRightarrow{}  (\mforall{}u,v:|r|.    (u  =  v  \mLeftarrow{}{}\mRightarrow{}  a  (u  +r  (-r  v)))))

Date html generated: 2017_10_01-AM-08_18_17
Last ObjectModification: 2017_02_28-PM-02_03_12

Theory : rings_1

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