Nuprl Lemma : fun-connected_antisymmetry
∀[T:Type]. ∀[f:T ⟶ T]. ∀[x,y:T]. (x = y ∈ T) supposing (x is f*(y) and y is f*(x)) supposing retraction(T;f)
Proof
Definitions occuring in Statement :
retraction: retraction(T;f),
fun-connected: y is f*(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
retraction: retraction(T;f),
exists: ∃x:A. B[x],
prop: ℙ,
fun-connected: y is f*(x),
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top
Lemmas referenced :
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
retraction-fun-path,
retraction_wf,
fun-connected_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
lemma_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination,
unionElimination,
imageElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T].
\mforall{}[x,y:T]. (x = y) supposing (x is f*(y) and y is f*(x)) supposing retraction(T;f)
Date html generated:
2016_05_15-PM-05_07_10
Last ObjectModification:
2016_01_16-AM-11_32_44
Theory : general
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