### Nuprl Lemma : connex_shift

`∀[A,B:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀[S:B ⟶ B ⟶ ℙ].`
`  ∀f:A ⟶ B. (RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f) `` Connex(B;x,y.S[x;y]) `` Connex(A;x,y.R[x;y]))`

Proof

Definitions occuring in Statement :  rels_iso: `RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f)` connex: `Connex(T;x,y.R[x; y])` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` connex: `Connex(T;x,y.R[x; y])` rels_iso: `RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f)` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` or: `P ∨ Q`
Lemmas referenced :  connex_wf rels_iso_wf or_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution hypothesisEquality cut lemma_by_obid isectElimination thin sqequalRule lambdaEquality applyEquality hypothesis functionEquality cumulativity universeEquality dependent_functionElimination addLevel orFunctionality productElimination independent_functionElimination

Latex:
\mforall{}[A,B:Type].  \mforall{}[R:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[S:B  {}\mrightarrow{}  B  {}\mrightarrow{}  \mBbbP{}].
\mforall{}f:A  {}\mrightarrow{}  B.  (RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f)  {}\mRightarrow{}  Connex(B;x,y.S[x;y])  {}\mRightarrow{}  Connex(A;x,y.R[x;y]))

Date html generated: 2016_05_15-PM-00_03_34
Last ObjectModification: 2015_12_26-PM-11_24_48

Theory : gen_algebra_1

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