Nuprl Lemma : member-rat-complex-boundary
∀k:ℕ. ∀K:ℚCube(k) List. ∀f:ℚCube(k).
((f ∈ ∂(K))
⇐⇒ (∃c:ℚCube(k). ((c ∈ K) ∧ (↑Inhabited(c)) ∧ f ≤ c ∧ (dim(f) = (dim(c) - 1) ∈ ℤ))) ∧ (↑in-complex-boundary(k;f;K)))
Proof
Definitions occuring in Statement :
rat-complex-boundary: ∂(K),
in-complex-boundary: in-complex-boundary(k;f;K),
rat-cube-dimension: dim(c),
inhabited-rat-cube: Inhabited(c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
l_member: (x ∈ l),
list: T List,
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rat-cube-sub-complex: rat-cube-sub-complex(P;L),
rat-complex-boundary: ∂(K),
bfalse: ff,
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
face-complex: face-complex(k;L),
all: ∀x:A. B[x]
Lemmas referenced :
istype-nat,
filter_wf5,
member_filter,
member-face-complex,
member-rat-cube-faces,
in-complex-boundary_wf,
istype-assert,
l_member_wf,
nil_wf,
subtract_wf,
int_subtype_base,
istype-int,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
equal-wf-base,
rat-cube-face_wf,
subtype_rel_list,
rat-cube-faces_wf,
eqtt_to_assert,
inhabited-rat-cube_wf,
list_wf,
map_wf,
concat_wf,
rc-deq_wf,
rational-cube_wf,
remove-repeats_wf
Rules used in proof :
promote_hyp,
dependent_pairFormation_alt,
independent_pairFormation,
independent_functionElimination,
dependent_functionElimination,
sqequalBase,
equalityIstype,
universeIsType,
productIsType,
setIsType,
equalitySymmetry,
equalityTransitivity,
rename,
setElimination,
addEquality,
natural_numberEquality,
minusEquality,
intEquality,
productEquality,
setEquality,
applyEquality,
independent_isectElimination,
productElimination,
equalityElimination,
unionElimination,
inhabitedIsType,
lambdaEquality_alt,
because_Cache,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
sqequalRule,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}K:\mBbbQ{}Cube(k) List. \mforall{}f:\mBbbQ{}Cube(k).
((f \mmember{} \mpartial{}(K))
\mLeftarrow{}{}\mRightarrow{} (\mexists{}c:\mBbbQ{}Cube(k). ((c \mmember{} K) \mwedge{} (\muparrow{}Inhabited(c)) \mwedge{} f \mleq{} c \mwedge{} (dim(f) = (dim(c) - 1))))
\mwedge{} (\muparrow{}in-complex-boundary(k;f;K)))
Date html generated:
2019_10_29-AM-07_58_46
Last ObjectModification:
2019_10_21-AM-10_11_37
Theory : rationals
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