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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