Nuprl Lemma : meq_transitivity
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y,z:X]. (x ≡ z) supposing (x ≡ y and y ≡ z)
Proof
Definitions occuring in Statement :
meq: x ≡ y,
metric: metric(X),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
meq: x ≡ y,
metric: metric(X),
implies: P ⇒ Q,
prop: ℙ,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
guard: {T},
trans: Trans(T;x,y.E[x; y]),
all: ∀x:A. B[x]
Lemmas referenced :
meq-equiv,
req_witness,
int-to-real_wf,
meq_wf,
metric_wf,
istype-universe
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
applyEquality,
setElimination,
rename,
natural_numberEquality,
independent_functionElimination,
universeIsType,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality,
productElimination,
dependent_functionElimination
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[x,y,z:X]. (x \mequiv{} z) supposing (x \mequiv{} y and y \mequiv{} z)
Date html generated:
2019_10_29-AM-10_56_16
Last ObjectModification:
2019_10_02-AM-09_37_28
Theory : reals
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