### Nuprl Lemma : monot_shift

`∀[A,B:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀[S:B ⟶ B ⟶ ℙ].`
`  ∀opa:A ⟶ A. ∀opb:B ⟶ B. ∀f:A ⟶ B.`
`    RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f) `` monot(B;x,y.S[x;y];opb) `` monot(A;x,y.R[x;y];opa) `
`    supposing fun_thru_1op(A;B;opa;opb;f)`

Proof

Definitions occuring in Statement :  rels_iso: `RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f)` monot: `monot(T;x,y.R[x; y];f)` fun_thru_1op: `fun_thru_1op(A;B;opa;opb;f)` 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 :  so_apply: `x[s1;s2]` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` fun_thru_1op: `fun_thru_1op(A;B;opa;opb;f)` implies: `P `` Q` monot: `monot(T;x,y.R[x; y];f)` rels_iso: `RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f)` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` guard: `{T}`
Lemmas referenced :  monot_wf rels_iso_wf fun_thru_1op_wf iff_transitivity iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation cut introduction sqequalHypSubstitution isect_memberEquality isectElimination thin hypothesisEquality axiomEquality hypothesis rename applyEquality lemma_by_obid lambdaEquality functionEquality cumulativity universeEquality independent_functionElimination independent_pairFormation dependent_functionElimination productElimination because_Cache equalityTransitivity equalitySymmetry independent_isectElimination

Latex:
\mforall{}[A,B:Type].  \mforall{}[R:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[S:B  {}\mrightarrow{}  B  {}\mrightarrow{}  \mBbbP{}].
\mforall{}opa:A  {}\mrightarrow{}  A.  \mforall{}opb:B  {}\mrightarrow{}  B.  \mforall{}f:A  {}\mrightarrow{}  B.
RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f)  {}\mRightarrow{}  monot(B;x,y.S[x;y];opb)  {}\mRightarrow{}  monot(A;x,y.R[x;y];opa)
supposing  fun\_thru\_1op(A;B;opa;opb;f)

Date html generated: 2016_05_15-PM-00_03_46
Last ObjectModification: 2015_12_26-PM-11_24_54

Theory : gen_algebra_1

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