Nuprl Lemma : member-rat-complex-subdiv-sub-cube
∀k,n:ℕ. ∀K:n-dim-complex. ∀c:ℚCube(k). ((c ∈ (K)') ⇒ (∃a:ℚCube(k). ((a ∈ K) ∧ rat-sub-cube(k;c;a))))
Proof
Definitions occuring in Statement :
rat-complex-subdiv: (K)',
rational-cube-complex: n-dim-complex,
rat-sub-cube: rat-sub-cube(k;a;b),
rational-cube: ℚCube(k),
l_member: (x ∈ l),
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rational-cube-complex: n-dim-complex,
prop: ℙ,
subtype_rel: A ⊆r B
Lemmas referenced :
member-rat-complex-subdiv2,
member-rat-cube-complex-inhabited,
l_member_wf,
rational-cube_wf,
rational-cube-complex_wf,
is-half-cube-sub-cube,
rat-sub-cube_wf,
istype-assert,
is-half-cube_wf,
rat-complex-subdiv_wf,
istype-nat
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
dependent_pairFormation_alt,
independent_pairFormation,
isectElimination,
independent_isectElimination,
universeIsType,
setElimination,
rename,
sqequalRule,
productIsType,
because_Cache,
inhabitedIsType,
independent_functionElimination,
applyEquality,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}k,n:\mBbbN{}. \mforall{}K:n-dim-complex. \mforall{}c:\mBbbQ{}Cube(k).
((c \mmember{} (K)') {}\mRightarrow{} (\mexists{}a:\mBbbQ{}Cube(k). ((a \mmember{} K) \mwedge{} rat-sub-cube(k;c;a))))
Date html generated:
2020_05_20-AM-09_23_25
Last ObjectModification:
2019_11_14-PM-08_23_48
Theory : rationals
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