Nuprl Lemma : ss-fun-fun
∀[X,Y,Z:SeparationSpace]. ss-homeo(X ⟶ Y ⟶ Z;X × Y ⟶ Z)
Proof
Definitions occuring in Statement :
ss-homeo: ss-homeo(X;Y),
ss-fun: X ⟶ Y,
prod-ss: ss1 × ss2,
separation-space: SeparationSpace,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
ss-function: ss-function(X;Y;f),
ss-eq: x ≡ y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
pi1: fst(t),
pi2: snd(t),
or: P ∨ Q,
ss-point: Point(ss),
record-select: r.x,
ss-fun: X ⟶ Y,
set-ss: {x:ss | P[x]},
mk-ss: Point=P #=Sep cotrans=C,
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
fun-ss: A ⟶ ss,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
ss-ap: f(x),
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
prod-ss: ss1 × ss2,
rev_uimplies: rev_uimplies(P;Q),
ss-homeo: ss-homeo(X;Y),
cand: A c∧ B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
separation-space_wf,
prod-ss-point,
pi1_wf_top,
ss-point_wf,
ss-function_wf,
pi2_wf,
prod-ss-sep,
ss-sep_wf,
ss-fun-eq,
subtype_rel_self,
ss-fun_wf,
set_wf,
sq_stable__ss-eq,
equal_wf,
ss-eq_transitivity,
not_wf,
or_wf,
prod-ss_wf,
exists_wf,
ss-fun-point,
ss-fun-sep,
ss-eq_weakening,
ss-eq_wf,
ss-ap_wf,
all_wf,
eta_conv,
squash_wf,
true_wf,
pair_eta_rw,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
lambdaEquality,
sqequalRule,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
applyEquality,
setElimination,
rename,
hypothesisEquality,
productElimination,
independent_pairEquality,
setEquality,
functionEquality,
productEquality,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
inlFormation,
independent_isectElimination,
inrFormation,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
dependent_pairFormation,
unionElimination,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality,
universeEquality,
natural_numberEquality,
instantiate
Latex:
\mforall{}[X,Y,Z:SeparationSpace]. ss-homeo(X {}\mrightarrow{} Y {}\mrightarrow{} Z;X \mtimes{} Y {}\mrightarrow{} Z)
Date html generated:
2020_05_20-PM-01_20_02
Last ObjectModification:
2018_07_07-AM-01_55_27
Theory : intuitionistic!topology
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