### Nuprl Lemma : quotient-of-quotient

`∀T:Type. ∀R:T ⟶ T ⟶ ℙ.`
`  (EquivRel(T;x,y.x R y)`
`  `` (∀Q:(x,y:T//(x R y)) ⟶ (x,y:T//(x R y)) ⟶ ℙ`
`        (EquivRel(x,y:T//(x R y);u,v.u Q v) `` u,v:x,y:T//(x R y)//(u Q v) ≡ x,y:T//(x Q 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` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` so_lambda: `λ2x y.t[x; y]` prop: `ℙ` subtype_rel: `A ⊆r B` so_apply: `x[s1;s2]` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` istype: `istype(T)` ext-eq: `A ≡ B` and: `P ∧ Q` quotient: `x,y:A//B[x; y]` infix_ap: `x f y` guard: `{T}`
Lemmas referenced :  equiv-on-quotient quotient_wf infix_ap_wf subtype_rel_dep_function subtype_quotient equal-wf-base quotient_subtype_quotient subtype_rel_self equiv_rel_wf quotient-member-eq
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_functionElimination hypothesis isectElimination because_Cache sqequalRule Error :lambdaEquality_alt,  instantiate cumulativity universeEquality applyEquality Error :inhabitedIsType,  independent_isectElimination functionEquality functionExtensionality Error :universeIsType,  independent_pairFormation pointwiseFunctionalityForEquality pertypeElimination productElimination equalityTransitivity equalitySymmetry Error :equalityIsType1,  productEquality Error :functionIsType,  hyp_replacement

Latex:
\mforall{}T:Type.  \mforall{}R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}.
(EquivRel(T;x,y.x  R  y)
{}\mRightarrow{}  (\mforall{}Q:(x,y:T//(x  R  y))  {}\mrightarrow{}  (x,y:T//(x  R  y))  {}\mrightarrow{}  \mBbbP{}
(EquivRel(x,y:T//(x  R  y);u,v.u  Q  v)  {}\mRightarrow{}  u,v:x,y:T//(x  R  y)//(u  Q  v)  \mequiv{}  x,y:T//(x  Q  y))))

Date html generated: 2019_06_20-PM-00_33_09
Last ObjectModification: 2018_09_30-PM-00_36_21

Theory : quot_1

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