### Nuprl Lemma : usym_shift

`∀[A,B:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀[S:B ⟶ B ⟶ ℙ].`
`  ∀f:A ⟶ B`
`    ((∀[x,y:A].  R[x;y] supposing R[x;y])`
`    `` RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f)`
`    `` UniformlySym(B;x,y.S[x;y])`
`    `` UniformlySym(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)` usym: `UniformlySym(T;x,y.E[x; y])` uimplies: `b supposing a` 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` usym: `UniformlySym(T;x,y.E[x; y])` rels_iso: `RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f)` member: `t ∈ T` prop: `ℙ` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` so_lambda: `λ2x.t[x]` uimplies: `b supposing a` subtype_rel: `A ⊆r B` so_apply: `x[s]` sq_stable: `SqStable(P)` iff: `P `⇐⇒` Q` and: `P ∧ Q` squash: `↓T` rev_implies: `P `` Q` guard: `{T}`
Lemmas referenced :  usym_wf rels_iso_wf uall_wf isect_wf subtype_rel_self uimplies-iff-squash-implies
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution applyEquality hypothesisEquality cut introduction extract_by_obid isectElimination thin sqequalRule lambdaEquality hypothesis instantiate universeEquality because_Cache functionEquality cumulativity productElimination independent_functionElimination imageMemberEquality baseClosed imageElimination dependent_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
((\mforall{}[x,y:A].    R[x;y]  supposing  R[x;y])
{}\mRightarrow{}  RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f)
{}\mRightarrow{}  UniformlySym(B;x,y.S[x;y])
{}\mRightarrow{}  UniformlySym(A;x,y.R[x;y]))

Date html generated: 2019_10_15-AM-10_32_24
Last ObjectModification: 2018_08_25-PM-05_13_09

Theory : gen_algebra_1

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