Nuprl Lemma : mcompact_functionality_wrt_homeomorphic+
∀X,Y:Type. ∀d:metric(X). ∀d':metric(Y). (homeomorphic+(Y;d';X;d) ⇒ mcompact(X;d) ⇒ mcompact(Y;d'))
Proof
Definitions occuring in Statement :
mcompact: mcompact(X;d),
homeomorphic+: homeomorphic+(X;dX;Y;dY),
metric: metric(X),
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
homeomorphic+: homeomorphic+(X;dX;Y;dY),
exists: ∃x:A. B[x],
and: P ∧ Q,
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
mcompact: mcompact(X;d),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q,
prop: ℙ,
mcomplete: mcomplete(M),
mk-metric-space: X with d,
metric: metric(X),
so_lambda: λ2x.t[x],
so_apply: x[s],
mfun: FUN(X ⟶ Y),
mcauchy: mcauchy(d;n.x[n]),
compose: f o g,
nat: ℕ,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
mconverges: x[n]↓ as n→∞,
mconverges-to: lim n→∞.x[n] = y,
m-unif-cont: UC(f:X ⟶ Y),
rless: x < y,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
sq_stable: SqStable(P)
Latex:
\mforall{}X,Y:Type. \mforall{}d:metric(X). \mforall{}d':metric(Y).
(homeomorphic+(Y;d';X;d) {}\mRightarrow{} mcompact(X;d) {}\mRightarrow{} mcompact(Y;d'))
Date html generated:
2020_05_20-PM-00_01_24
Last ObjectModification:
2019_12_04-PM-06_54_06
Theory : reals
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