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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