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
Home
Index