### Nuprl Lemma : all_rng_quot_elim_a

`∀r:CRng. ∀p:Ideal(r){i}.`
`  ((∀x:|r|. SqStable(p x))`
`  `` (∀d:detach_fun(|r|;p)`
`        ∀[F:|r / d| ⟶ ℙ]. ((∀w:|r / d|. SqStable(F[w])) `` (∀w:|r / d|. F[w] `⇐⇒` ∀w:|r|. F[[w]{|r / d|}]))))`

Proof

Definitions occuring in Statement :  quot_ring: `r / d` ideal: `Ideal(r){i}` crng: `CRng` rng_car: `|r|` detach_fun: `detach_fun(T;A)` type_inj: `[x]{T}` sq_stable: `SqStable(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  type_inj: `[x]{T}`
Lemmas referenced :  all_rng_quot_elim
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep cut lemma_by_obid hypothesis

Latex:
\mforall{}r:CRng.  \mforall{}p:Ideal(r)\{i\}.
((\mforall{}x:|r|.  SqStable(p  x))
{}\mRightarrow{}  (\mforall{}d:detach\_fun(|r|;p)
\mforall{}[F:|r  /  d|  {}\mrightarrow{}  \mBbbP{}]
((\mforall{}w:|r  /  d|.  SqStable(F[w]))  {}\mRightarrow{}  (\mforall{}w:|r  /  d|.  F[w]  \mLeftarrow{}{}\mRightarrow{}  \mforall{}w:|r|.  F[[w]\{|r  /  d|\}]))))

Date html generated: 2016_05_15-PM-00_24_36
Last ObjectModification: 2015_12_27-AM-00_00_20

Theory : rings_1

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