Nuprl Lemma : strict-fun-connected_transitivity2
∀[T:Type]. ∀f:T ⟶ T. (retraction(T;f) ⇒ (∀x,y,z:T. (y is f*(x) ⇒ z = f+(y) ⇒ z = f+(x))))
Proof
Definitions occuring in Statement :
retraction: retraction(T;f),
strict-fun-connected: y = f+(x),
fun-connected: y is f*(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
strict-fun-connected: y = f+(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
guard: {T},
prop: ℙ,
not: ¬A,
false: False,
uimplies: b supposing a
Lemmas referenced :
fun-connected_transitivity,
not_wf,
equal_wf,
fun-connected_wf,
retraction_wf,
fun-connected_antisymmetry
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
productEquality,
cumulativity,
functionExtensionality,
applyEquality,
functionEquality,
universeEquality,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
voidElimination,
independent_isectElimination,
equalityTransitivity
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T. (retraction(T;f) {}\mRightarrow{} (\mforall{}x,y,z:T. (y is f*(x) {}\mRightarrow{} z = f+(y) {}\mRightarrow{} z = f+(x))))
Date html generated:
2016_10_25-AM-11_04_15
Last ObjectModification:
2016_07_12-AM-07_12_20
Theory : general
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