Nuprl Lemma : homeomorphic_weakening
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. (X ≡ Y ⇒ (dX = dY ∈ metric(X)) ⇒ homeomorphic(X;dX;Y;dY))
Proof
Definitions occuring in Statement :
homeomorphic: homeomorphic(X;dX;Y;dY),
metric: metric(X),
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
mfun: FUN(X ⟶ Y),
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
is-mfun: f:FUN(X;Y),
all: ∀x:A. B[x],
so_apply: x[s],
squash: ↓T,
prop: ℙ,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
homeomorphic: homeomorphic(X;dX;Y;dY),
exists: ∃x:A. B[x],
cand: A c∧ B,
meq: x ≡ y,
metric: metric(X)
Lemmas referenced :
subtype_rel_weakening,
meq_wf,
squash_wf,
true_wf,
metric_wf,
subtype_rel_self,
iff_weakening_equal,
is-mfun_wf,
ext-eq_inversion,
metric-on-subtype,
ext-eq_wf,
istype-universe,
meq-same,
req_witness,
int-to-real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
hypothesisEquality,
applyEquality,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
independent_isectElimination,
sqequalRule,
universeIsType,
imageElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
productElimination,
independent_functionElimination,
universeEquality,
dependent_pairFormation_alt,
productIsType,
functionIsType,
setElimination,
rename,
equalityIstype,
independent_pairFormation
Latex:
\mforall{}[X,Y:Type]. \mforall{}[dX:metric(X)]. \mforall{}[dY:metric(Y)]. (X \mequiv{} Y {}\mRightarrow{} (dX = dY) {}\mRightarrow{} homeomorphic(X;dX;Y;dY))
Date html generated:
2019_10_30-AM-06_23_34
Last ObjectModification:
2019_10_02-AM-10_30_59
Theory : reals
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