Nuprl Lemma : rat-cube-third-not-in-face
∀[k:ℕ]. ∀[p:ℝ^k]. ∀[c:ℚCube(k)].
  ∀f:ℚCube(k). (¬in-rat-cube(k;p;f)) supposing ((¬(f = c ∈ ℚCube(k))) and f ≤ c) supposing rat-cube-third(k;p;c) ∧ (↑Inh\000Cabited(c))
Proof
Definitions occuring in Statement : 
rat-cube-third: rat-cube-third(k;p;c)
, 
in-rat-cube: in-rat-cube(k;p;c)
, 
real-vec: ℝ^n
, 
nat: ℕ
, 
assert: ↑b
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
and: P ∧ Q
, 
equal: s = t ∈ T
, 
inhabited-rat-cube: Inhabited(c)
, 
rat-cube-face: c ≤ d
, 
rational-cube: ℚCube(k)
Definitions unfolded in proof : 
req_int_terms: t1 ≡ t2
, 
rdiv: (x/y)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
nequal: a ≠ b ∈ T 
, 
int_nzero: ℤ-o
, 
sq_type: SQType(T)
, 
exists: ∃x:A. B[x]
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
decidable: Dec(P)
, 
uiff: uiff(P;Q)
, 
less_than': less_than'(a;b)
, 
less_than: a < b
, 
rneq: x ≠ y
, 
rat-interval-third: rat-interval-third(p;I)
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
guard: {T}
, 
squash: ↓T
, 
true: True
, 
subtype_rel: A ⊆r B
, 
nat: ℕ
, 
prop: ℙ
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
top: Top
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
or: P ∨ Q
, 
rat-point-interval: [a]
, 
rat-interval-face: I ≤ J
, 
rational-interval: ℚInterval
, 
real-vec: ℝ^n
, 
rat-cube-face: c ≤ d
, 
in-rat-cube: in-rat-cube(k;p;c)
, 
rational-cube: ℚCube(k)
, 
rat-cube-third: rat-cube-third(k;p;c)
, 
and: P ∧ Q
, 
false: False
, 
implies: P 
⇒ Q
, 
not: ¬A
, 
all: ∀x:A. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
Lemmas referenced : 
real_term_value_add_lemma, 
real_term_value_const_lemma, 
real_term_value_var_lemma, 
real_term_value_mul_lemma, 
real_term_value_sub_lemma, 
real_polynomial_null, 
int-rinv-cancel, 
rmul-rinv3, 
radd_functionality, 
req_transitivity, 
req_weakening, 
req_functionality, 
req-iff-rsub-is-0, 
rsub_wf, 
req-implies-req, 
req-rat2real, 
nequal_wf, 
int_formula_prop_wf, 
int_term_value_mul_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_eq_lemma, 
int_formula_prop_not_lemma, 
istype-int, 
intformeq_wf, 
intformnot_wf, 
full-omega-unsat, 
decidable__equal_int, 
int_subtype_base, 
subtype_base_sq, 
itermAdd_wf, 
rinv_wf2, 
itermConstant_wf, 
itermVar_wf, 
itermMultiply_wf, 
itermSubtract_wf, 
rmul_preserves_req, 
rless_wf, 
rless-int, 
int-to-real_wf, 
rmul_wf, 
radd_wf, 
rdiv_wf, 
req_fake_le_antisymmetry, 
istype-universe, 
equal_wf, 
iff_weakening_equal, 
subtype_rel_self, 
real_wf, 
true_wf, 
squash_wf, 
rational-interval_wf, 
rat2real_wf, 
rleq_wf, 
rat-interval-third_wf, 
istype-nat, 
real-vec_wf, 
rational-cube_wf, 
inhabited-rat-cube_wf, 
istype-assert, 
rat-cube-third_wf, 
rat-cube-face_wf, 
in-rat-cube_wf, 
int_seg_wf, 
rat-interval-face_wf, 
istype-void, 
pi1_wf_top, 
rationals_wf, 
pi2_wf, 
in-rat-cube-face
Rules used in proof : 
int_eqEquality, 
sqequalBase, 
dependent_set_memberEquality_alt, 
dependent_pairFormation_alt, 
approximateComputation, 
intEquality, 
cumulativity, 
inrFormation_alt, 
closedConclusion, 
promote_hyp, 
independent_pairFormation, 
universeEquality, 
instantiate, 
baseClosed, 
imageMemberEquality, 
imageElimination, 
productIsType, 
isectIsTypeImplies, 
functionIsType, 
functionIsTypeImplies, 
because_Cache, 
rename, 
setElimination, 
natural_numberEquality, 
equalitySymmetry, 
equalityTransitivity, 
equalityIstype, 
universeIsType, 
voidElimination, 
isect_memberEquality_alt, 
independent_pairEquality, 
lambdaEquality_alt, 
sqequalRule, 
applyLambdaEquality, 
unionElimination, 
inhabitedIsType, 
applyEquality, 
dependent_functionElimination, 
functionExtensionality, 
hypothesis, 
independent_isectElimination, 
hypothesisEquality, 
isectElimination, 
extract_by_obid, 
independent_functionElimination, 
productElimination, 
sqequalHypSubstitution, 
thin, 
lambdaFormation_alt, 
cut, 
introduction, 
isect_memberFormation_alt, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}].  \mforall{}[p:\mBbbR{}\^{}k].  \mforall{}[c:\mBbbQ{}Cube(k)].
    \mforall{}f:\mBbbQ{}Cube(k).  (\mneg{}in-rat-cube(k;p;f))  supposing  ((\mneg{}(f  =  c))  and  f  \mleq{}  c) 
    supposing  rat-cube-third(k;p;c)  \mwedge{}  (\muparrow{}Inhabited(c))
Date html generated:
2019_11_04-PM-04_43_32
Last ObjectModification:
2019_11_04-PM-03_32_34
Theory : real!vectors
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