### Nuprl Lemma : all_rng_quot_elim

`∀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]))))`

Proof

Definitions occuring in Statement :  quot_ring: `r / d` ideal: `Ideal(r){i}` crng: `CRng` rng_car: `|r|` detach_fun: `detach_fun(T;A)` 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 :  all: `∀x:A. B[x]` implies: `P `` Q` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` member: `t ∈ T` crng: `CRng` rng: `Rng` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` sq_stable: `SqStable(P)` ideal: `Ideal(r){i}` detach_fun: `detach_fun(T;A)` quot_ring: `r / d` rng_car: `|r|` pi1: `fst(t)` quot_ring_car: `Carrier(r/d)` quotient: `x,y:A//B[x; y]` squash: `↓T` infix_ap: `x f y`
Lemmas referenced :  rng_car_wf all_wf quot_ring_wf rng_subtype_rng_sig crng_subtype_rng subtype_rel_transitivity crng_wf rng_wf rng_sig_wf detach_fun_properties squash_wf equal-wf-base assert_wf rng_plus_wf rng_minus_wf quot_ring_car_subtype sq_stable_wf detach_fun_wf ideal_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis independent_functionElimination applyEquality instantiate independent_isectElimination sqequalRule lambdaEquality because_Cache dependent_functionElimination pointwiseFunctionalityForEquality pertypeElimination productElimination equalityTransitivity equalitySymmetry imageMemberEquality baseClosed productEquality functionEquality cumulativity universeEquality

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]))))

Date html generated: 2018_05_21-PM-03_14_41
Last ObjectModification: 2018_05_19-AM-08_07_56

Theory : rings_1

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