### Nuprl Lemma : quotient-squash

`∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ].  x,y:T//E[x;y] ≡ x,y:T//(↓E[x;y]) supposing EquivRel(T;x,y.E[x;y])`

Proof

Definitions occuring in Statement :  equiv_rel: `EquivRel(T;x,y.E[x; y])` quotient: `x,y:A//B[x; y]` ext-eq: `A ≡ B` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` squash: `↓T` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` implies: `P `` Q` ext-eq: `A ≡ B` and: `P ∧ Q` subtype_rel: `A ⊆r B` quotient: `x,y:A//B[x; y]` all: `∀x:A. B[x]` squash: `↓T` prop: `ℙ`
Lemmas referenced :  equiv_rel_wf equal-wf-base quotient-member-eq squash_wf quotient_wf equiv_rel_squash
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality independent_functionElimination hypothesis independent_pairFormation pointwiseFunctionalityForEquality independent_isectElimination pertypeElimination productElimination because_Cache dependent_functionElimination equalityTransitivity equalitySymmetry imageMemberEquality baseClosed productEquality universeEquality imageElimination independent_pairEquality axiomEquality isect_memberEquality functionEquality cumulativity

Latex:
\mforall{}[T:Type].  \mforall{}[E:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    x,y:T//E[x;y]  \mequiv{}  x,y:T//(\mdownarrow{}E[x;y])  supposing  EquivRel(T;x,y.E[x;y])

Date html generated: 2016_05_14-AM-06_08_07
Last ObjectModification: 2016_01_14-PM-07_33_19

Theory : quot_1

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