Nuprl Lemma : homeomorphic_transitivity
∀[X,Y,Z:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[dZ:metric(Z)].
(homeomorphic(X;dX;Y;dY) ⇒ homeomorphic(Y;dY;Z;dZ) ⇒ homeomorphic(X;dX;Z;dZ))
Proof
Definitions occuring in Statement :
homeomorphic: homeomorphic(X;dX;Y;dY),
metric: metric(X),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
homeomorphic: homeomorphic(X;dX;Y;dY),
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
compose: f o g,
cand: A c∧ B,
all: ∀x:A. B[x],
meq: x ≡ y,
metric: metric(X),
mfun: FUN(X ⟶ Y),
prop: ℙ,
sq_stable: SqStable(P),
is-mfun: f:FUN(X;Y),
so_apply: x[s],
squash: ↓T,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
compose-mfun,
req_witness,
int-to-real_wf,
meq_wf,
compose_wf,
mfun_wf,
homeomorphic_wf,
metric_wf,
istype-universe,
sq_stable__meq,
meq_functionality,
meq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
applyEquality,
setElimination,
rename,
natural_numberEquality,
independent_functionElimination,
universeIsType,
independent_pairFormation,
productIsType,
functionIsType,
inhabitedIsType,
instantiate,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination
Latex:
\mforall{}[X,Y,Z:Type]. \mforall{}[dX:metric(X)]. \mforall{}[dY:metric(Y)]. \mforall{}[dZ:metric(Z)].
(homeomorphic(X;dX;Y;dY) {}\mRightarrow{} homeomorphic(Y;dY;Z;dZ) {}\mRightarrow{} homeomorphic(X;dX;Z;dZ))
Date html generated:
2019_10_30-AM-06_24_13
Last ObjectModification:
2019_10_02-AM-10_43_04
Theory : reals
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