Nuprl Lemma : xxorder_functionality_wrt_breqv
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ]. ((R <≡>{T} R') ⇒ (order(T;R) ⇐⇒ order(T;R')))
Proof
Definitions occuring in Statement :
xxorder: order(T;R),
binrel_eqv: E <≡>{T} E',
uall: ∀[x:A]. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
xxorder: order(T;R),
uimplies: b supposing a,
uiff: uiff(P;Q)
Lemmas referenced :
xxorder_wf,
binrel_eqv_wf,
xxrefl_functionality_wrt_breqv,
xxtrans_functionality_wrt_breqv,
xxanti_sym_functionality_wrt_breqv,
xxrefl_wf,
xxtrans_wf,
xxanti_sym_wf,
binrel_eqv_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
addLevel,
productElimination,
independent_functionElimination,
because_Cache,
independent_isectElimination,
levelHypothesis,
promote_hyp,
andLevelFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. ((R <\mequiv{}>\{T\} R') {}\mRightarrow{} (order(T;R) \mLeftarrow{}{}\mRightarrow{} order(T;R')))
Date html generated:
2016_05_15-PM-00_01_21
Last ObjectModification:
2015_12_26-PM-11_26_22
Theory : gen_algebra_1
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